This won't fly
What you have proposed is not to Code, due to the current loop created by the "solo" hot wire going up to the fan then returning back via the neutral going down to the other switch. This violates the intent of the 300.3(B)/310.10(H) prohibitions against current loops and paralleled wires which can create magnetic fields and ensuing ...
It’s hard to tell for sure but if it was working normally until recently I’d say the electronics for the remote control receiver are failing.
You can bypass the remote unit and connect one of the wall switches directly to the light kit. You can keep using it that way or replace the remote and receiver with a universal ceiling fan remote control kit.