If you want to save this tree, forget about getting any fruit this year. I would cut the trunk at about 3 ft off the ground and try to get laterals to develop from that point.
Then keep it pruned to an open vase system.

Unless you can figure out how to get more leaves on that tree it's worthless. It won't mature decent fruit regardless of the variety.

It could be a seedling, therefore always give poor quality fruit no matter what you do. Is there any sign of a graft? Having no visible graft is not definitive, it might be below the ground. But if not is visible I would assume seedling and act accordingly. Actually, I would start a makeover immediately either way.

you don't show your zone but in my area we are still good to cut it off at 3' and do a total make-over... except you probably don't have scions. So that puts you back to making cuts to for new growth low and bud graft to those in the fall, or with scions in the spring. If you have never grafted before, start reading about it now

"It could be a seedling, therefore always give poor quality fruit no matter what you do. Is there any sign of a graft?"

The vast majority of seedling peaches will produce pretty good peaches. Peaches aren't a fruit like apples where they've been changed so radically that you can get basically a crabapple.

No signs of any graft. Its hard to see growth in the picture because of the background and there are lots of dead twiggy branches. I think it probably has never been treated for peach leaf curl and that probably killed a lot of stuff. It was pretty covered in leaves last fall and I didn't spray it at all over the winter. I'm just learning this stuff now

So should I wait a couple years until my other trees start producing and then cut it off at 3' and graft on some better varieties?

Cutting it to 3' will likely kill the tree unless there are leaves at that point- peaches don't tend to have dormant buds along old wood ready to spring to life. I do see some leaves that might be low enough and you can probably get away with cutting just above them.

I don't really think such a tree is worth the effort. I've seen plenty of seedling trees with terrible to mediocre fruit so it is a bit of a crap shoot- you said it is clingstone and if it was true to seed it would by all odds be free. The fact that it is clingstone is an indication of low quality from my experience with seedlings.

You won't be able to graft anything on it until next year and by the time the tree provides crop your other peach trees will be doing so as well- peaches don't take long. However, it might be a fun project for you.

Just thought I would show an update of what this tree looks like. It sprouted 2 limbs at the perfect height to be used as nurse branches

If you want to save this tree, you already have scaffolds developing at a good spot. If you can get scion wood of something you like, why not top graft it right above the new scaffolds and keep them also. Then you'd have a 2&1 tree. If you don't like the old variety, you can always prune it out.

That's what I plan to do. I want to graft on 3 varieties. I bought 2 of the varieties as potted trees a couple weeks ago. I still need to locate the 3rd one. It doesn't really matter yet though. I don't think I'm going to do it this spring, I think I'm going to wait another year.