Playing at high levels is bad practice.
Most people don't really seem to appreciate that every time you play, you're putting in a bit of practice. Reinforcing the habits you have, good or bad, and deepening the rut in the road that your wagon travels in.
When you're trying to create a cool character, it's only too easy to fall into the trap of thinking that the stronger the character is, the cooler they are. At some point, you might end up playing a world-shattering demigod, reveling in just how much fictional power your character has, while asking for the rest of the group to be shocked and amazed by how cool your character is.
The sad part is, that's actually the easiest way to make a character no one likes. A strong character that lacks a personality beyond "I'm strong" and who has obtained their power solely because the narrative demanded it is the recipe for a forgettable midboss, placed into the position of one of the main characters only because you happen to play him.
Playing at high levels is a bad idea, but not for any of the reasons you say. You're wrong about pretty much everything. The idea that you need to be high-level to revel in PC power or that players are going to start doing that when they level up is utterly terrible.
Playing at high levels is a bad idea because designers don't test high-level stuff. The game breaks down really fast. As an example that I'm sure everyone remembers, D&D 3rd edition went to print despite breaking down at high levels because the higher-level stuff wasn't well-tested and most groups don't get past level 6 anyway. For an example that old-timers like me recall, older editions had avatars of gods with like 90 hp. You were supposed to kill them then retire to go be a king somewhere or some shit, crowned by your own hand Conan-style because you were just that hard. You weren't actually supposed to keep playing past level 9, past that Thar Be Dragons...I mean utterly untested content!
You know what my take on high-level play is? Things fall apart because it becomes impossible for people to take everything into account. At low levels, the players and the DM are going to be on the same page. The DM knows what the PC can do and what they should want to do, and the players have a good general idea of the threats and challenges ahead of them. But at higher levels, there's a chance of divergence. Our last campaign was very underwhelming because the DM basically expected us to be up to speed with the powers of enemies we'd never seen before (in and out of character) while most of us were RPing characters who weren't exactly into party cohesion (bunch of random people forced to run errants for vastly more powerful people).
Low-level stuff basically plays itself, but high-level stuff requires both more effort in preparation and socially. The best quest I've ver run as a DM was the very first quest of the campaign, and for the reasons I mentioned: Things were clear. I could anticipate what people would do, it was a closed environment, and they did not have unpredictable powers. And because money didn't matter a whole lot, the party actually cared about the nuances of the setting.
I agree with one of OP's points in a way, high level play does hurt new players trying to make cool and memorable characters. A lot of new players tend to equate "Cool power = cool character" and then get confused when the other party members don't seem to care. I've lost count of the number of new players that when asked about their character only describe what they can do with the enthusiasm of someone that just figured out that caps locks is cruise control for cool and it saddens me to no end.
Well they don't really see it the other way since they're still new. They think that if a character can't even fuck up a Orc that they won't even survive long enough to be memorable.
The problem is a good RPG is about a journey typically, from being noobs to people who can possibly get digits to a goddess.
Higher levels are hard to get to because campaigns don't persist to. It's totally fine to can a campaign at a level and start their next campaign fresh at that level.
You can still do a lot with enemies at higher levels, its the challenge of making those enemies do what they are supposed to and not just stat wise. I cite Tucker's Kobolds on this one. But frequently everyone forgets that the flavor text has a point.
I have not got past level 5 in D&D when I was a player in 7 when I DM'ed. I am starting my next campaign at 5, high levels for D&D 5e don't seem as bad in power creep but don't have as much fun to them as older editions or so I have been told.
If you use an alternate level scaling system, it may be better.
For example; if every level each character only gets a small bonus to character power, and at level 20 they have moderate bonuses but they still aren't demigods, I think that doesn't quite equate to 'high level player' as most people think it. At least that's how I see it.
The only real problem in playing at high levels is that it is much harder to challenge players and stay inside setting logic at the same time.
Let's take an example (actually, one of mine campaigns). Players start out as adventuring group, who hunts particularly nasty magic artifact to destroy it. They found out that evil wizard found it, and, comparing power of the artifact and his own, plans to start war between two kingdoms and rule that's left of them. Players disrupt his plans, kill his minions first, then go for the wizard. They successfully defeat him, yet artifact eludes them once again: it's creator, evil deity, stoles it from the wizard at the worst possible time for him (fighting the party), and it's servants return said artifact to his temple. Characters track down the artifact and the temple, storm it with an entire army, and in the climatic battle slay both the deity's high priest and it's avatar. Artifact destroyed, hurray for heroes.
Now, after this campaign they certainly were high-level heroes (although we weren't using the class-based system). The fighter could take on entire squad of trained warriors and take them all out, the priest was regularly called on his deity personally (and had become saint of his own religion), the swashbuckler-sorcerer had became a hurricane of magically-enchanced stabs, and shapeshifter could turn into a dragon.
I wanted to close the game and run something different, because characters kinda hit the ceiling - there were very little things in the setting which could hurt them. But players got really tied up to charcters after half-year of regularly gaming, and wanted to continue playing.
So I had to find way to make adventure continue, and I have had to make it challenging. But it wasn't easy, given that they became paragons of their area. So, I created a plot, there evil deity and demon king conspire to open the gate to demon dimension and destroy the world.
After that, everything went downhill quickly.
To players, it was still pretty much fun, but I had to practically ruin my setting with my own hands, constantly and subtly changing it so that characters weren't the biggest fish in the pond. After having demon king lieutenants utterly stomped into the ground (the one died before he even had the chance to react), I somewhat overreacted and made the demon king a fucking nightmare. That followed was the goriest TPK in my RPG gaming, and totally destroyed relations with players, who said that was so uneven that they didn't had a chance. Strictly, that wasn't truth, but balancing encounters at highest levels can be somewhat tricky, and I really made a mistake.
If every adventure you do is a D&D dungeon stomp, then yeah high-level play is going to be complete ass. Either because you run out of things that can give an adventuring party which has survived that long a good challenge ("What, another dragon? Pffft!"), because the differences in power levels just become too great because the checks and balances start breaking down hard, or simply because the setting doesn't properly accomodate high-level characters. I mean look at something nice and generic like Greyhawk, there really isn't any room for a party of level-20 adventurers to run around and actually have adventures. But if you're playing a system that isn't D&D, or at least playing a campaign that isn't about who can kill a horde of demiliches the fastest, that doesn't need to be a problem.
I'm personally not a fan of campaigns that start at a high level, because to me that just feels like reading only the last chapter of a book. It lacks the proper build-up, the natural growth that makes the end result all the more valuable. Just rolling up a level 20 character is infinitely more boring than starting out as a level 1-3 nobody and taking it from there. It's the same reason why I never plan out my character builds in advance. Someone who, at level one, has already planned out what his character will be like at level 20 is without exception a bad roleplayer. They not only neglect the entire journey that brings their character to level 20, they're also playing a game where "getting really strong" is the main goal they've set for themselves from the start, like it's some kind of natural reward for putting in enough time.
However, none of that means playing at a high level is bad. It just means that playing at a high level requires more effort. It requires a GM who knows what he's doing and can keep the game's setting interesting even when all mundane challenges are no longer challenging. Which either means coming up with better, more fantastic challenges, or running a game that isn't all about the dice. It also requires players that are more interested in playing their character than in winning the game. If you're at high level, the game has already been "won". At that point the party can probably take over a kingdom single-handedly and live the rest of their lives in comfort and indulgence, able to do pretty much whatever they want. So it becomes more important than ever to actually think about what it is your character wants in the first place. You're no longer taking orders from the local guild, or taking jobs killing goblins in order to afford some cool new equipment. The world is already yours for the taking, so what are you going to do with it?
>The world is already yours for the taking, so what are you going to do with it?
It feels like it's easier to answer that question if you're playing a campaign where characters begin as lower level, local heroes and progress their way through to higher level killing machines. Hopefully, by the time they've got there, who there characters are, what they care about and the world they're inhabiting has been so well fleshed out that the question "what do you do now?" doesn't take long to answer.
I'd want to encourage my players to keep a notebook full of grudges and the people who have wronged their characters, so that we can have an epilogue to the campaign where a bunch of vindictive level twenty murderhobos descend on the people who wronged them. The miller who short-changed the party on their first rat-killing-in-the-basement job. The warlike royalty of the neighbouring kingdom who ordered a scorched earth policy as they retreated back to their own kingdom. The corrupt priest who turned a blind eye to the cultists using the church basement. Now, it's theoretically possible that the characters (or players) might not hold a grudge, and maybe they might even go around forgiving the people who wronged them, but those odds are pretty slim.