I don't have any experience with persimmons but I have heavily pruned back Apples to graft in new varieties. It seems like the established root system directs a lot of energy to the new graft and promotes rapid new growth to replace the portion of the tree that has been removed. I use cleft grafts on <2" diameter branches or I prune back to a small side branch where I can do a whip and tongue graft.
I wouldn't cut the tree to the ground. If you have scaffold branches at 3 - 4' I think you could cut them back to 1' long and graft a scion on each scaffold for rapid conversion to your new persimmon variety.
I have grafted persimmons; the difference I noticed was that it took most of the summer before they began growing(?). I used 3 whip & tongue grafts on 3, year old "American" rootstocks.
I now have 4 Asian persimmon trees; I dug out my "5th," an American persimmon, as it was so brittle it would loose branches in a summer breeze! Personally, I wouldn't graft onto your existing tree; I suspect 'takes' would be iffy, or inconsistent, and with the extremely brittle nature of persimmons, I wouldn't trust the resulting structure.
You could cut it to the ground, causing a multitude of sprouts, all very good candidates for grafting. But there'd likely be a lot of dieback on the root system (again), and you'd eventually want to select only one grafted shoot as your main leader...
Personally, I'd dig it out and replace it with whatever I wanted. Grafted trees are more costly than ever, and Persimmons seem to demand a premium, but wasted years add up too. Dubyadee's recommendations are sound, you could go that route if you're so inclined (adventurous). Me, I'd just get out the catalogs ... and my favorite shovel 
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