Seems to be some PGA238 (socket 2) on eBay. Does it need to be ZIF 237?

I see it, but that's not the ZIF one I had in mind. But that's a good point. If there's some other ZIF socket that fits, I can use that. Does PGA237 go by some other name like PGA 19x19?

Haha, I found and bought some! They're socket2, but just as good for my museum purposes.
https://w.componentonline.com/product/425C-238B/916591-1 (beige)
https://w.componentonline.com/product/425C-238/916501-1 (blue)

How much did they end up wanting for these ? The white ones look nice, with tabs for heatsink retention clip even.

It's going to depend on how many you order and your negotiating (you specify your bid price in the request for quote). I ordered 8 + 4 for $120.

Now, I'm going to design and CNC cut some pockets to house them in

That is not so bad then, I will keep this place in mind because there are some 486 boards I'd like to use some better sockets on, and these ones look really good and especially because of the heatsink clips.

You know how to properly desolder those sockets? I see you've made circuit boards. What's the secret? I tried with a heat gun to the back while prying each side with a 1mm thick steel strip. Look what happened.

Heatgun things properly until all the solder has melted, including the pins connecting to internal layers (this can take a while), and only then try to lift the socket out. If you have a preheater, this will help a lot.

It is quite likely all the capacitors nearby will go bad afterwards, from all the excess heat they'll be getting. It is also advisable to remove all the jumpers etc. since they may deform and get stuck on the pins.