Seeing as there is no mold, it suggests that it dries out rather quickly, but something looks like it gets or got wet there, so may want to look real hard as to what or if the what has already been fixed. If those are gas lines that probably go up under a cabinet there might be a sink drain pipe that is under the floor of the cabinet but above that flooring, and could possibly only be getting wet periodically, or could something be splashing up rather often?
The idea you have has some merit but could be more of a Band-Aid depending on how well you can do it (I'm more of a carpenter and not really a trailer repairman), but because of the missing layers of existing plywood there might be space between your new layer of plywood and the existing plywood, which may leave some sponginess but would keep you from falling through. I would probably try to fill the area smoother with some floor patch (a product from HD or Lowes) to take up the space before covering with more plywood. Do make sure that your 2x4 reinforcement is mounted/bolted to something substantial, and not just screwed up to the existing plywood or it will rip right out. Might want to consider angle iron in place of 2x4's or at least those "Strong-Tie" 2x4 hangers to help secure them.
The proper way to fix would still be to remove what's above and replace but obviously that could be expansive and possibly cost prohibitive on an old trailer. That said I would probably try the route you're looking at too.
Good luck.
