I've always ridden my rear tires until the cords start showing, which means virtually all the tread rubber is gone, not just a little chunk. A little random chunk missing or a cut doesn't mean much IMO if the cords are intact and not compromised. Cut/damaged cords would normally be evidenced by bulging. The fact that this happened on a trainer is a bit curious. It suggests perhaps that the tread rubber and/or it's bond to the cords has been been compromised by the stresses involved. Having big chunks or strips of tread rubber coming off while riding would be less than ideal.

For typical cuts or divots in the tread, I just leave them be.