Thanks for the replies on this issue. I have a couple two year old peaches (Redhaven and Coralstar) in the ground with similar problems (too many scaffolds). They are starting to bud swell now with this milder weather. I’ll keep these suggestions in mind.
Would it be OK to prune it now, or wait until the really cold weather has passed?
I think pruning right before bloom is said by some to stimulate flowering. If so, that’s no good until freezes have passed. You’ll want the peach to stay dormant for a while longer in a best-case scenario.
Pruning peach trees after bloom hasn’t been a problem for me, even when wanting them to grow taller. They do that fine anyway 
Oh, and wood heals faster on them when it’s warmer, so it might even be a good idea to prune after winter anyway, imho.
Leave anything you will be grafting on until you graft, btw. Or cut it some but not as far back as for grafting. You want a fresh cut the day of grafting.
Ok, thanks, I can wait until later. It’s just Feb 27th, so it’s still way early. We’re supposed to get some temps down in the mid to upper 20s over the next week anyways.
Yes, a failed graft is not the end of the branch. Their should be plenty left! Do another there the next year, and the year after that!
You can do them now or anytime in the spring or summer. Experts are heading toward later pruning. What we have been doing is late winter/early spring (now) pruning for shape, and summer pruning for size. Some changing thought in recent years. First it was only cherry, now apricot has been added for summer pruning only due to certain infections. You want a dry spell after you prune so do not prune before rain is expected.
You are such a good coach. Thanks. These are great things to know.