I have seen worse PCB's repaired. But it's only when the ability to replace the entire PCB isn't an option and parts are obsolete. (I am into retro computers, so there's a lot of cannibalizing and reviving old computers) But that stuff is typically only two layer PCBs. That's relatively new enough to be up to a 4 layer board. It certainly has vias for 2 at least. It's hard to make out, but it might be a 3 or 4 layer board. So that makes things even more complicated to repair. The first thing to repair something like this would be to put it in a ultrasonic cleaner. Then break out the magnifying glasses to check for any top layer breaks. When a trace is broken, you can repair it with a bodge/flying lead. Or sometimes simply scraping off the insulator and bridging the break can work.
Looking at the board, it's mainly passive components. The TLE6220GP is what's driving the solenoids in the transmission. The rub is the firmware in that CPLD, PCS doesn't appear to have released the 2100 schematic. So it's also unlikely you'll get a copy of their firmware either. It has a JTAG header on it. But I'd also bet they have the security bit set to it. The only way you'd get the firmware is to try to glitch it and dump the firmware from it. (And that's a maybe)
Could it be repaired? Sure. Would it be cost effective? Not a chance. Would it be fast? Also not a chance. Would it work? A slim chance.
I'd imagine that *some could be repaired. I bet the 6220 drivers go out on them a lot or other passives. But when it comes to any of their "proprietary" IP, you're most likely not going to fix it.