A relatively minor complaint I have about 5th edition is having to wait until third level to get my subslcass. In a few of the D&D campaigns I've run, players have wanted to skip to level three because "that's when it gets fun." One of the reasons we wait until third level to get our subclass is to avoid encouraging players to multiclass by dipping their toes into various classes to get those abilities at level one.
But multiclassing is an optional rule. Why build character generation and progression around an optional rule? Let's just have our subclass at first level and if that makes multiclassing too powerful then don't allow that as an option.
The actual problem here is not multiclassing, though that is a symptom of it.
It is the idea that 1st level (and somewhat 2nd) must be simultaneously:
1. The level at which brand-new players begin the game, and thus have a gentler introduction to mechanics
2. The "hardcore survival" level, where HP are scarce and monster damage is very high
3. A solid but incomplete foundation of what is to come
Being pulled in three different directions causes serious design consequences, one of them being what you speak of here. This is why robust "novice level" rules would be a significantly better approach. They would, I admit, require the design effort to make them in the first place, so that is a non-negligible cost. But once you have done that, all of the downstream problems evaporate. The group served by point 2 now has a near-endless playground, especially if the rules incorporate ideas like 13th Age's "incremental advance" rules, allowing DMs to spool out levelling almost indefinitely. The brand-new players served by point 1 can have tailored structures fit for their needs, given a bigger cushion of HP than currently exists, while still having slimmed-down mechanics so they aren't overwhelmed by choices instantly. And point 3 ceases to be necessary, because now level 1 can be a full and complete foundation.
Everyone wins, except the designers who have to put in the work to make more rules. Admittedly, I'm not a designer here, so I'm asking someone other than me to do work for my benefit....but as a (potential) paying customer, I see that as entirely appropriate.
This is exactly the sort of problem you get when you design based purely on bellyfeel and design-by-committee (or, in this case, design-by-poorly-structured-survey), rather than having a cohesive design goal. I had highlighted this (and given my feedback about it) all the way back in D&D Next; the problem's always been there in 5e.
i dislike multiclassing because it's a hideously clunky implementation and causes class design to need to be far too careful about potentially unexpected combinations, though i'm more than fine with subclasses all being at 3rd,
but if it was removed i would still desire for there to be a better system to combine archetypes, perhaps if subclasses were built to be more interchangeable and feats contained bigger design space to pick up individual pieces from other classes.
so i could build an Oath of Vengance Rogue and pick up Rage as a feat, or a Circle of the Land Sorcerer with a Pact of the Chain familiar.
This is a game design nightmare and essentially guaranteed to result in at least one of two major problems. On the one hand, outright useless or massively OP subclasses because you stapled Circle of the Moon (all about shapeshift) to Monk (a class that can't shapeshift) or Oath of Vengeance (powerhouse Paladin spells) onto Warlock (auto-scaling spell slots) or the like. Features never tested together now intersect, with combinatoric explosion ensuring it isn't remotely feasible for the designers to test everything, and becomes even less so with every published book. The char-op potential skyrockets, generally producing even more incentive to scour the rules and franken-build the best possible result.
On the other hand, perhaps as an effort to avoid the above...mass blandification. Circle of the Moon can't do anything particularly dramatic to Wild Shape, because
anyone can take it and only Druids have that feature. Or, you make it so everyone can have every class feature if they invest into it, thus completely eliminating the concept of class fantasy and turning D&D into a full (albeit patchy/chunky) point-buy game. Pretty sure that would go over like a lead balloon, so I think it very unlikely. Either way, you "solve" the problem above by making it so it really doesn't matter very much what class you've picked, which flies in the face of data I've seen indicating that players mostly think of their characters class-first rather than species-first; species is what you visualize, but class is what you
do, and the doing is extraordinarily important.
I guess there is a third approach, but it would mean not actually doing the thing you've described here. Instead, you give every subclass restrictions that must be met, so Circle of the Moon can only be taken by Druids. This makes things massively more complicated on the players' end without really changing any of the balance issues, unless the restrictions are coupled with major testing...which I don't think they will be.