Well, the woods available to me locally do come in 1 1/2, 2" or thicker.

If I did have a reasonable bandsaw with a riser block, could it resaw a
10" or 12" piece with decent accuracy?

Jack

If its set up properly and has a sharp blade. I resaw frequently however,
you still need to plane or run the pieces thru a drum sander after resawing
them.

Rewsawing will save you money, particularly when the primary thickness you
use is 1/2". Its a horrible waste to plane a 3/4" board down to 1/2" when
you have 2" wood available to resaw.

Dave