It is hard to tell if the floor has rot or if that is just mold.
If this is just mold confined to the top layer of your osb and the osb seems structurally sound I'd probably:
- wear a mask, turn the fan on, get a bucket of bleach water and some shop paper towel, wear gloves, aggressively scrub the area with shop towels. I'd probably get a plastic wire brush that is used to clean floors and do that as the second pass.
- dry the area, use a heater and or dehumidifier to dry the area after you've cleaned it.
- get a wire brush and brush any areas that still look like they need more work, sweep it up and bag it.
- paint the area with a mold killing primer/sealer. killz or zinsser.
- tile, use epoxy grout, use a silicon caulk to seal the tile to the tub.
Mold itself isn't dangerous it is the airborne mold spores that aren't good for you. If you did nothing and sealed it in the floor it would likely be fine as well. The concern with that is that the thin set wouldn't properly bond to the area with the layer of mold.
Also note mold in that bathroom is probably the least worrisome as you spend the least amount of time in that room.
Ripping the tub out would make that job 3x harder for little real gain. If the floor has significant rot then you cut the piece of flooring out, add nailers and patch it. I can't imagine pulling the tub unless the joists under the subfloor were significantly rotten - even then I'd sooner pull the ceiling below (assuming the tub is otherwise nice and the tile job is good).