I have a couple of quilt tops that my mom had worked on 30 years ago. She tore a lot of the fabric, even when she wanted 2.5" sashing.
I will tear to get a straight edge, but then normally work off that and don't tear more. I do tear when working with extra wide backing, but that normally tears very well. I tear off the selvages before putting it on the frame too.
Once I get it mounted, I'll roll it back and forth a few times with a bit of tension on it, and that will straighten it out quite nicely. I always pre wash, and sometimes it's quite off before it gets mounted.