It seem that Slackware and Steam don't always play well together at the moment.
The other day I had problems with CS:GO getting stuck, and now I have found EAC games don't work, and also don't give a helpful error message. This is what I get whenever trying to start an EAC game:

And looking in the Proton prefix AppData I find some more messages:

[19:24:05:526] [Windows] [EAC Launcher] [Info] Launcher finished with: 210, 'Unexpected error. (#1)'.

and

[2023.01.26-19.24.04] [Connection] Connect result: No error (0) Response Code: 200 Destination IP: 18.244.179.51
[2023.01.26-19.24.04] Download Progress: 100
[2023.01.26-19.24.05] [EAC Callback] Code: 509. Message: 'Unexpected error. (#1)'.

So apparently there is No error but there is an error?

This problem is probably caused by incompatible versions of libraries in Slackware at the moment. I had thought about how I could test with different versions of glibc (which is what I thought might be the problem) but found Conty instead which makes things much easier.
Just download the latest conty.sh from the releases pages and run it with the steam option

$ chmod +x conty.sh
$ ./conty.sh steam

Then Steam will run using the version of the Arch Linux libraries that are included in the Conty image. This seems to have fixed EAC for me, and also made the main Steam UI feel a bit smoother for some reason - that must be something to do with Chrome.