Except you're a wizard, which means you can learn all of the spells. You can decide to be the second guy when you complete your next long rest, and then be a third guy (with powers like the second one, but different effects on the next day. If you have one class which can cast all of the spells, then that class replaces all of the other classes that can each only cast some of the spells. And once you've played one wizard, you've played them all.
To contrast, if you had twelve distinct wizard classes and they each had their own spell lists, then you could play twelve different characters before things started getting repetitive.
Perhaps in theory but not in practice. Divine casters are the ones who can actually cast all the spells with a day's rest, and they have been so since 3e. The wizard gets one or two spells per level, plus whatever the DM is willing to give him on top of that. Even if some Monty Haul DM were to give you access to the entire wizard list, copying them all into your spellbooks would require a wealth of both gold and time. Plus, you'd need a portable hold to tote your library around with you.
Despite the fact that divine casters
literally have access to all divine spells, I haven't have any difficulty keeping my divine characters from being repetitive throughout the editions, but that's just my personal experience.
Arguably the only edition that made no-magic a practical option, at all, so far. 5e just needs some more varied and flexible non-magical PC choices, though, and it could get there. It's already OK with relatively few magic items. HD, Second Wind, and sub-classes like the BM and Mastermind establish the sort of mechanics that could do that, it just needs a lot more on a more flexible class chassis....And the Monk. But you couldn't really play with just those classes: the need for healing and other spellcasting resources was just too great. Actually /zero/ classes. Variety consisting of Tanky DPRx3, DPR&skills x2. That's a very little variety. While you can get some between-combat healing from HD, lessening the need for the traditional 'Band-aid' Cleric, you still need in-combat healing and the many other contributions that only casters can make. 5e's very open about that with it's blurb on magic, early on - but it has room to grow into supporting such playstyles.
I agree that 4e was the best at handling a no-magic campaign, although I do think it was possible in other editions to varying degrees. 5e would be the next best IMO, and I'd also agree that it has room to grow in achieving its full potential in this respect.
I didn't count monk because I don't consider it a no-magic class, given that it's always had a mystical aspect to it. Admittedly, this might not be an issue in a no-magic campaign that allows psionics, where you can attribute the monk's mystical powers to psionic ability.
I disagree with you on there being zero classes in 5e that are non-magical. Sub-classes ought to be considered separately from the classes, as you can easily ban a sub-class without impacting the class. The base Barbarian, Fighter, and Rogue classes have zero magical abilities. If you ban the Totem Warrior, Eldritch Knight, and Arcane Trickster, you still have 3 fully viable and playable classes.
I don't think that magic was especially rare in AD&D. But its use was more differentiated: clerics and druids generally used it less in combat than did wizards; illusionists didn't have access to direct damage (until Chromatic Orb in UA); paladins were clearly magical/supernatural (with their aura of protection, LoH etc) but in combat fought using steel and sinew.
Differences in spell list, and in distribution of spell functions, help achieve this.
In 4e there is the combination of different power lists (though less distribution of function than in AD&D, I would say) and greater limitation on re-training (once per level rather than after every long rest). So any given two PCs are perhaps likely to feel more different.
I don't have enough of a sense of 5e class design to know which (if any) of these approaches might be the way to try and reduce a sense of ubiquity/homogeneity.
Different spell lists might help. For myself, I prefer having options. One illusionist might focus on pure illusions like the 1e Illusionist. Another might mix illusory walls of fire with real ones to keep enemies on their toes. I think that more options gives a class greater replay value and makes it more interesting in general.