Yep, too much of the traces are gone and while one can rewire for sound, it's very hard to restore button contacts. You're better off asking for a replacement clean controller board from someone who has a Turboexpress (or PC Engine GT) that are for parts.
If someone has the experience and time, maybe build a new PCB board in CAD software like Eagle, and then submit it to a PCB fab to get a new board made, transfer the chip and the ribbon cable, and put in a new cap and parts plus new audio plug and drop it in the TE. I can't imagine any demand for replacement controller board since normally they don't fail.
The main board seems fine. It seems like the battery leak didn't go up to the main board. I do suggest not removing the main board, the cable that connects to cart socket is fragile and can break if you shifted the board the wrong way.