I used to think it stalled sometimes but I think it's actually Safari. When I quite Safari and re-launch it, the problem goes away. This is with keeping Private Relay on the whole time. It just seems like Safari on desktop and Private Relay sometimes don't get along, oddly.
I also have not experienced any issues with Private Relay on any of my iOS devices.
Be that as it may however, the issue never happens with Private Relay off so who knows? I wish the issue didn't happen.