The main knock on the WalMart "Schwinns" seems to be cheap drivetrain and very poor assembly. Drivetrain issues are minimal with a singlespeed and taking the bike out of the store in its box will negate the big problem.

What's wrong with Grandma's bike? The lifespan of those old Sturmey Archer 3 speeds is +/- forever. Unless it's catastrophic frame your $100 should be more than enough to get it running.