I think I've figured it out! It isn't the yarn quality that is the main issue. It's the connection between the tool's hook and latch.
My yarn was fraying so much that I slowed right down to look carefully at what the problem was. When pulling the yarn through, the yarn strand was somehow being pierced by the hook instead of wrapped around it, shredding the yarn as the stitch was finished.
I think that a loose or skewed connection between the end of the hook and the end of the latch means that the yarn gets split when it is pushed past the connection. Perhaps there are high-quality tools that never do that, but not mine!
My solution is to push the yarn right up into the hook, past the connection, after closing the latch but BEFORE pulling it through the canvas. It's not foolproof, and it's slow, but it's working much better now.