If you give it a good cleaning, and then reflow the solder on the corroded connections with lots of flux before you turn it on, it might be good enough.
Be sure to blow out any cruft from under the chips that are soldered on as well. Heating up corroded pins on chips and reflowing the solder will help too.
If you're creative and handy you might be able to fix the clips on those broken CPU and I/O board connectors on that board. You'd need to figure out what shape they are by extracting one of the old ones, and cutting a new one, then carefully inserting the new one in. Just be careful to not allow it to get loose and short out a pin next to it.
If the corrosion made it all the way inside the traces inside the board, you'd have to to use a VOM set to continuity testing and beep out the connections, should a connection not work, you'd have to add a jumper wire and glue it to the board. But then, over time, it'll just keep eating traces and it will fail again.
You wouldn't need super specialized equipment to desolder, just a desoldering sucker with a soldering iron. There are cheap ones that have both the pump and a soldering iron together like this for under $10:
https://www.amazon.com/Velleman-VTDESOL3U-Vacuum-Desoldering-Heater/dp/B00B88FRME/You'd still be doing it single pin at a time. The only danger is from overheating a pin and lifting a pad, or damaging the pin on the connector, if you're not careful enough.
Give it a shot, if it's already dead, you don't have much to lose.
John used to have the fully populated motherboards over at Vintage Micros, maybe reach out to him and see if he has any left if you don't have the time to invest and just want it to work.
Edit: When my MBP17" died with a GPU failure, I bought one of these for ~$40:
https://www.amazon.com/Kohree-Rework-Station-Solder-Digital/dp/B00JVM3WBC/ - it's basically a hot air gun whose temperature you can precisely control. You could use this along with a lot of liquid flux to either reflow solder on the back of the board, or even to help desolder the board. Be careful with it to avoid melting the board or components. I'd look up the temperature of leaded solder, such as is used in the Lisa first and set it to that.
In my case I used this to reflow the GPU in my old 17" macbook pro as I said, so if you've got some old Macs that might have that same issue, it might be worth investing in. Would work on other things where the BGA connectors can fail, like Sony Playstations. I've used this thing 3x already, so it more than paid for itself by getting another 2 years out of an otherwise dead machine.
I'd be careful to not overheat the solder joints if you're just reflowing them with this thing, otherwise, they'd melt and fall through the other side. So you'd want a quick pass once or twice, just enough to make the solder go liquid but not more than that.
I'd practice on some old electronics junk first like old PC boards or radios or whatever that are broken and you don't care about.