My own experience with t shirt quilts as a top or the back does not work well to use as a binding. You need something that will help to hold the quilt in place.

I'd suggest using a cotton for the binding. It helps to keep everything in place.

Makes great dust rags.

I would think you could use that for binding but I would cut it the length of the back instead of crosswise. Crosswise seems to have more stretch. I wouldn't interface it as that would add bulk that could make it difficult. Using woven fabric would be more stable but, since you are hoping to be frugal, using the backs of the shirts would be worth a try.

I have used leftover t-shirt scraps and sewn to base for new puppy pads. Work well. Wash well.
I never have big pieces left because I use backs as sashing for the actual fronts.
I have also usd some larger pieces to make pillow case for quilt or smaller pillows to match the quilt

At one time, we could buy t-shirt knits at the Munsingwear Remnant Room that was in Minneapolis - long closed now. I had a pretty good stash of knits, too.