One stat to rule them all, or what if we merged Dex and Str?

[diplomancer]

Again, why would EVERY very nimble waif have this mysterious reserve of strength they suddenly call upon to lift the 1000 lb. door (or whatever), but couldn't before, while the clumsy waifs don't have this ability?

I can think of several explanations. I believe the easiest one is "she could do it before, she just didn't know it, having never tried; after all, she had a pretty developed musculature, being able to do back flips and all that sort of things. Add to that the adrenaline of the moment, and it's perfectly understandable".

The capacity to lift heavy objects or wield a two-handed sword as well as you do a dagger - better, even, since the unified stat means the bigger damage die is probably better unless you've got a LOT going for making that dagger amazing - is there if you've got unified Str and Dex, and "role playing" that you don't have it means, no, you don't "adrenaline surge" to massive strength in those clutch moments, but rather that you pretend you fail on checks you should be succeeding.

Well, regardless, if they're trying the checks, they are going to succeed at them.

"The strong guy" and "the nimble guy" are two distinct archetypes

Agreed

and need distinct mechanics to represent them

This is the point of this thought experiment. To check whether this is true; or, to be more precise, whether they can be represented by other mechanics than the ability score. Perhaps you can just represent them by giving the "strong guy" Athletics, Heavy Armor, and a Maul and the "nimble guy" Acrobatics, Light Armor and a Rapier? Just not having proficiency in Athletics (if you want your character to be more nimble than strong) will go a long way.

"Role playing" that you don't have the mechanical abilities your sheet says you do is akin to deciding that, no, your warlock doesn't recover spells on a short rest because his patron is stingy. You can do it, but your really RPing a delusional character who thinks they lack an ability they have. Or you're otherwise ignoring or breaking the rules. Doing so "for RP" to make yourself weaker is not much better than doing so "for RP" to make yourself stronger. I'm sure some DMs and tables will be okay with it, just as some are okay with breaking the rules "for RP" even if it makes you stronger in clutch situations, but if you're having to effectively house rule a weakness into your character because of a proposed change, that proposed change might be the problem.

I don't know about your sheets, but I believe the official sheets don't even have a "maximum encumbrance" field.

And I think it's pretty bad form to tell other players what their characters can do (unless they are newbies, and even then it'd be polite to wait for them to ask for your help). Specially if, the way a character is described, the other characters would have no idea that she's actually capable of doing that stuff.