And this, right here, is the origin of a lot of people's opposing views on multi-classing.
You
don't have to "do for all." It is
entirely possible and acceptable to say certain multiclass combos are acceptable and some aren't. You can have a pre-written list or even (horrors!

) decide case-by-case.
Or "I don't care for multi-class, and it's rare in my setting, so I'm only allowing one. You guys can decide who it is, or I can pick randomly from those who want it."
As long as you're not playing favorites with
players, these are no more unreasonable--and in many cases more so--than banning MCing completely.
Well, yes and no. I'd be more than happy to make unilateral case-by-case decisions. But I've found, over the years, that it avoids a lot of argument and bad feeling when everyone is on the same page and knows what to expect...and new people can be told by any/everyone else in the group "Yeah that's cool." or "No that's not how we do."...for pretty much any type of mechanics/rules question.
I agree with all of the above and in my homebrew game there is a very clear laundry list of what MC is permissible and what isn't, available and applicable to everyone [and NPCs], to maintain a pre-3e flavor. I imagine when I get to a "5e" game, we will keep the same parameters.
Since the OP was asking about the general 5e rule, in a kind of "yes or no/use 'em or don't" way, I didn't really think any of that kind of house rule consideration was needed. But sure, there are all sorts of limitations one can reasonably ascribe to the mechanics.
In terms of the on/off switch for the 5e MC rules, I meant the "do for one, do for all" as the players. If I allow the Warlock/Paladin/Sorcerer, then I have to allow the next guy who wants their Monk/Wizard/Ranger and the other who wants a basic Cleric/Fighter and so on...if they come to me with those. If all you have is the one crackpot, that you know will always come to you with some off the wall thing just because, then fine. But you can't allow one player to MC anything and then tell another player they can't MC however they want into their PC, too.
I mean, sure you
can. But it's not really nice or fair. So I wouldn't.
At the same time, I would think the best way to avoid any kind of confusion or even the appearance of favoritism, especially when dealing with strangers or new players to an established group or, I would guess, "organized play" is to make sure everyone is held to the same standard, i.e. allow/disallow, all classes, across the board, for everyone.