Some routers don't like iCloud Private Relay as it is a privacy measure. I have gotten notifications from some wireless networks that internet access is not available via iCloud Private Relay. I tell those networks (that want to track my every move) to fuck themselves.

It sounds like the switch is doing some type of management or traffic shaping that is breaking your internet connection. I know you said if it works out of the box as a dumb switch with no configuration but it seems obvious that it is doing something. Wi-Fi would bypass the switch unless you have a wireless access point plugged into the switch. My understanding is that Private Relay is basically Tor but through Cloudflare and other providers that Apple uses so it might be a good idea to Google Tor internet issues with a managed switch and see if anything there is applicable.

I’d also check through the settings of the switch and disable any settings you can that impact traffic flow in any way. If you have an actual dumb, unmanaged switch you can switch in, I’d try that too. Even if you have to buy one, they’re fairly cheap and can be a good tool to have in your back pocket to troubleshoot network issues.

Maybe i've missed it, but have you tried pluggin your macbook into the network without the switch? Switches are mostly level 2 network devices and sholdn't affect higher levels.

Does your switch have sticky Mac or any kind of port security turned on? It could be that private relay is randomizing your MAC address to prevent finger printing and it’s messing with the switches port-security settings or ARP table. Actually if that was the case then the local network shouldn’t work… do the other devices work on the switch while you have private relay on?