All editions had this problem.
The issue with high level D&D is that only spells scales well after ~13th level or so.
The following non-caster classes/subclasses in 5e continue to scale excellently:
The Fighter, who at 20th level is a chainsaw capable of attacking minimum of 8 times two rounds in a row twice per short rest. Assuming even a basic magic weapon (+1d6), the fighter can do damage equal to a failed save against the 1/day meteor swarm with each set of attacks. Some subclasses can do substantially better, and with a level-appropriate magic weapon such as a vorpal sword, the Fighter can expect to probably land an instant kill somewhere in those two rounds.
The Zealot barbarian, who becomes immune to death by hit point loss at 14th level and acquires an almost unbeatable grapple check.
The monk, who can turn invisible and gain resistance to all damage except force at 18th level, making himself an excellent infiltrator and off-tank, while also being one of the three best classes in the game for saving throws (alongside paladin and artificer).
All the other classes except rogue have spellcasting (spellcasting creep is a different topic).
As another specific example, rogues are the best generalist skill users, but I'm finding that the Battlemaster may not be far behind. I'm running a Battlemaster war leader in a campaign right now. At level 7 with Expertise from a feat, his Persuasion check is 1d20+8, + 1d8 from a maneuver. At level 20 with a Charisma of only 14, he'd be at 1d20+14 + 1d12 for a persuasion check range of 16-46. I'm not sure any spellcaster except Bard can keep up with this, and there are at least 6 skills Battlemasters can do this for. Even the 8th level spell Glibness doesn't do anything other than set a high floor for what the spellcaster's lowest roll can be.
Not every class scales as well to 20 (ranger), but most classes in 5e have pretty good scaling features. Having run high level several times... the non-casters are fine.