It's the walkers that generally open new paths to avoid muddy puddles in my experience. Not trying to say bikes are harmless to a forest, but likely less harmful than horses and less numerous than walkers - depending on location.
I had a go at a couple of MTBers riding through the ASSI (Area of special scientific interest) at the end of the road the other day. It doesn’t even go anywhere FFS!
I think the main paths are OK, but there are multiple puddles on them that have spread to cover the whole width because people skirt the sides.