FreeTheSlaves
Adventurer
Well, in that case the requirements should apply to single-class PCs too.
But that brings up the issue that 5E PCs are not actually all that dependent on their "prime stats." As others have already pointed out, you can be a devastatingly effective fighter with a Strength of 8; just pump your Dexterity and go the dual-wielding route. As long as you steer clear of spells with save DCs and Wisdom-dependent domain powers, your cleric will do fine with a crappy Wisdom. If you're a moon druid, you are actually well advised to pump Dexterity before Wisdom.
I think the 1st level class should be treated differently as it currently being done. The starting character is at least a young adult with a history captured through racial features, background and class choice. The next level gained through experience typically has nowhere near as much time spent before it occurs.
In the case of the Dex based 1st level fighter, I'd consider that more a perk of choosing Fighter at 1st level. Sure they've still got a Str save, exposure to heavy weapon & armour training, i.e. Fighter basics, but they've shot off on their specialization. The Rogue multi-classing into Fighter doesn't have the benefit of x years development. I'm not too concerned to see they effectively need to pump Str to gain proficiencies & features. Maybe they'll need to wait until after level 4, which is perfectly viable given the frequency of ability bumps and how the games seems to be built around a 16 prime stat.
As it stands Dex is simply waaay too good. Multi class restriction here and heavy armour negating it there have lowered its value slightly but it still needs a smack with the nerf hammer every time.
Even so, maybe Barbarian or Ranger are more suitable choices? Each has one requirement you probably already satisfy.
I agree that prime stats need a bit more empowering. My favourite class, the Paladin, seems to almost get more value from Con than Cha. For spell-casting I prefer the 10+spell level route and class features should draw on the stats a bit more. Maybe not so heavily as say 3E/4E, but just a touch more. Imo seeing a viable 9 Wis Cleric 10 is a bit of a travesty.
Idiot proofing doesn't need to be done too much throughout the game. The advice in the class descriptions is generally enough. However I do agree that the gloves should come on somewhat when it comes to multi-classing. I remember one player getting into his 3E Rogue/Sorcerer character but then the spells came slowly and weakly and the skills couldn't get enough development. Those edition specific problems may have been solved - though there's a question of 5E skills unresolved in my mind.