Replace subfoor to include the C channel is a must. A pain in the butt, maybe, but not that much more difficult than replacing what you have.
A shell off is not necessary if the frame is in good shape. I would get all the old sub floor out, including the C channel in stages as you replace the flooring. Then replace the two curved end pieces first with a full length piece that would end on a frame piece. I would half lap the end joint on the frame rail so that other subfloor pices lock into other pieces. Then I would cut pieces along the sides with again half lap joints over framework. With the final middle pieces to finish off.
https://www.amazon.com/CMT-835-502-1.../dp/B000P4LPJ6
Set up your router and router a bit of your subfloor material, then cut piece in two and place rabbit over rabbit; adjusting router depth until both pieces fit flush to each other. Now you have created a half lap joint.
Subfloor material of choice is Coosa board as it will not rot, grow mild, mildew or decay. Expensive, yes but you won't have to ever do this job again. I used 3/4" Coosa as 5/8" wasn't available up here and machined the edges down to 5/8" to fit channel.
Cheers
Tony