Sheldon Brown, IIRC, actually thinks Biopace rings are not that bad and he uses them on some of his bikes.
Go with the MTB frame. Of that vintage, it's rugged, and will last almost forever.
Rivetted chainrings means you won't be able to swap out individual rings -- the crank and rings are all rivetted together rather than having bolts that you can undo, and you have to buy the whole lot together. Fortunately, they are lower level Shimano MTB stuff such as Acera, and are cheap.
Be careful about swapping out cranks and BBs. Your BB is probably still cup-and-race with spindle. If you do go with a cartridge, you will need to ensure the chainline isn't radically altered, that the Q-factor is even on both sides (ie, the cranks/pedals are the same distance away from the seat tube, and that the smallest chainring has enough clearance on the chainstay. New cranks and chainrings will add another dimension you may have to take into account.
Your derailleur should be able to handle a moderate change in rings as it's already three-speed.
I suspect that you have a spin-on freewheel 7sp cluster on the rear. You may have to hunt around for one that meets the spec you want. I'm not sure they are available with an 11T cog. If you have a freehub set-up, you may be in a better position.
I'd probably go with a chainring upgrade for now, but with the caveats already mentioned.