there seems to be no trace of custom clients anywhere

Unsure if there are any clients fully supporting CPE like ClassiCube, but speaking about Minecraft Classic clients, they are enumerated here: https://wiki.vg/Classic_Client_List.

I have implemented my own Classic client called McLordClassic (link to GitHub repository), however it lacks a launcher that would make it easy to run the game. If you would like to try it and are comfortable with terminal, the command to start it is java -Xmx6G -XX:+UseG1GC -DmclordAddress=<server-address> -DmclordPort=<server-port> -DmclordUsername=<your-username> -DmclordMppass=<mppass> -jar desktop-1.0.jar. Also it requires McLordClassicPlugin in order to be able to connect to Classic servers.

Edited November 9, 2024Nov 9 by deewend

It looks like the clients on the list are abandoned & don't support classicube auth