I overcast the frayed edges with a zigzag stitch or the serger.
I have even sewn towels that have torn - like gotten caught on something - back together with a darning stitch.
I was raised "frugally".
I even - way back when - would cut worn washcloths down the middle and sew the less used outside edges together. Now these washcloths are in the rag pile for icky clean-ups.
If the overcasting is done soon enough with matching thread, it is barely noticeable.