This is the sort of stuff I do a lot -- other people's UFOs. Some are gifted to me, some are thrift store finds. There are times when I will remove seams to rescue just the fabric or rearrange the blocks, but not often. One (ok two) of my Tuesday Ladies goes to extreme lengths to rescue vintage tops.
Once it has been given to you it is yours to do with as you will! That can be "finishing" (which can be as intended or what you are willing to do) or re-use or passing on again. I know sooner or later I'll come across a big bunch of badly done 9-patches, I have an idea that involves slicing them and re-trimming but I don't want to do that to good blocks or blocks I made.
I recently posted some completed quilts, a fellow board member quilted them down. All of them came via donations/thrift store buys. While I put a couple of the tops together from the scraps, others were just added to/trimmed.
Quilted Tops -- but not by me!