a square inch is a square inch. it won't get bigger. if you try to squeeze more threads into it, the threads have to be thinner.
100% cotton 200 thread count sheets are fine. they're actually much stronger than fabric.
if you look at vintage or antique quilts that were actual utility quilts, you'll find that the fabric was relatively loosely woven by today's standards and yet they held up for over 100 years. anything too tightly woven will be punctured in the sewing and damaged. i'm sure you've seen holes in sheeting along the edges?
if you're hand-sewing, you need a needle that will slide between the fibres. i'm not a hand quilter, so i don't know what that would be, but it's important not to break or tear the threads.
if you do want to try sheeting, why not try a thrift store? a decent sheet, new, 100% cotton, flat, queen - king size, shouldn't cost more than $3 - 5. a good bargain.