I know you're coming into this more as an evangelist than anything else, but ... I disagree with this point.
Part of the joys of both OD&D and 1e are exactly the things you're disregarding here, at least IMHO.
Skill systems? Out of combat resolution? Others will disagree, but I don't really want a system for those things, if I'm running oldschool. Want to search the room? Tell me what you're doing. Don't want to give me details? I'll roll some d6's. Want to jump the chasm? DM fiat based on strength, class, armor, etc with maybe a die roll thrown in. And so on.
And as for all the miscellaneous sub-systems, that's part of the charm, too, imho. The more subsystems you have, the more modular a system is. Don't like the unarmed combat system? Well, who does? Plug in something else. Want to add weapon specialization? Go ahead. Want to change up the thief skill chart? Done!
Even one strength you mentioned - using any supplement for any edition - can work just dandy in 1e. It's no harder to add a feat-type system or a skill-type system if that's what you're looking for.
I know that C&C gets a lot of love from its fans, but it kinda leaves me cold. I find the SIEGE engine overrated & unnecessary. I don't particularly care for its use as a saving throw mechanic, either. But, if you love it, you can drop it into a 1e game with just as much ease as the other way around.
I just found C&C to be a strange half-step of a game, which tries to straddle the line between old-school and new-school, and in the process became charmless.
IMHO, YMMV, etc.
-O