Seems like a good plan. Could add bridging or blocking to help share the load (probably already has but if not add)
Can you probe the extent of the damage?
If so then maybe its worth dutching in a filler. That can be a lot of work but does mean the one complete joist isn't cut away.
If its a real wood (not ply etc) flooring then when you cutaway the original joist you'll have to cut the nails and no easy way to reinstall if its a modernish construction (ie subfloor).
But if the joist is gone in the pocket its doing no good.
