It's hard to follow what you want to do.

I think (may be wrong) the term nurse root grafting normally is grafting of a bit of root onto the bottom of a cutting. Or sometimes the grafting of an easy rooting type to the bottom of a hard-to-root type (the classic illustration in old horticultural literature is the latter, with two mulberries).

When the well-rooted cuttings are lifted, the bottom roots are sometimes intentionally cut off, otherwise "let the best roots win."

"Can I split a root stock in two and then graft the two root halves onto a scion? I want to bring an apple back to its true root."

I have a just a few apple scions and I want to hedge my propagation bets by doing numerous grafts. I thought I would commit one of three scions to being root-grafted.

I can get maybe three nurse-grafted pieces out of the one length of scion wood.

If I place the nurse-rooted pieces in bagged pots is there a good chance of success?

I want the apple on its true roots.

Look up the process known as stooling. It is a technique often used for generating rootstocks where a young tree is cut back severely,nearly to the ground. Sawdust or soil is mounded around the resulting shoots, and the next year they will have rooted and can be removed for further propagation.
I think you could do a standard whip graft on any rootstock (best a vigorous one) with your scion; and then plant it with the graft well below soil level. Let it grow out a year and then stool it the next spring. The resulting shoots should be self rooted version of your scion. and could be planted out the following fall or spring.

"If I place the nurse-rooted pieces in bagged pots is there a good chance of success?" answer: no.

"I have a just a few apple scions and I want to hedge my propagation bets by doing numerous grafts."..."I want the apple on its true roots."

answer:Graft the scions on to seedling root stocks. Plant the graft union below the soil line and the scion should grow its own roots in several years.