Attached are pics of the idler plate and the rotted plywood
The pictures are helpful.
The steering cables need to be replaced. They should be rust free and lightly lubricated with 30w oil. The rust will also abrade the sheaves on the idler pulleys. Inspect them to ensure the grooves are smooth and the axle round. The axles can develop a flat spot which makes them hard to turn.
Coosa would work, but be much more expensive than plywood and it is a bit harder to work with. A while back Andy Miller on Boatworks Today used it to rebuild a transom. The videos might give some insight.
Replacing on the damaged section might be more challenging than just replacing the whole piece. Trying to get a nice clean straight cut on the intact plywood with damaging the glass would difficult. The old plywood will come off pretty easily with a chisel and mallet, some 40 grit should take care of a lot the remaining wood.
The other issue is joint strength. Butt joints are weak because the bonding surface is small, that's why scarf joints are used. Trying to put a good scarf joint on a small piece of wood could be an exercise in frustration.
Ideally, the wood should be marine ply. Sourcing a small piece might be challenging. It might be possible to find some usable scrap at a boat yard. If not, the plywood should have multiple plies, the more plies the better. This makes it more dimensionally stable.