Interesting. Underlying network changes shouldn't make a difference to WG; TCIP routing is dynamic.
But you may be onto something about DNS. If, when you switch networks, the OS is overwriting the DNS server information that WG set up, that would do what you're describing. Restarting WG would re-assert the DNS serves that are configured. The one hitch is that normally this would only cause leakage, not failure to resolve... overwriting WG's DNS servers with public ones should still work.
Still, it's a good intuition, and if it were me, that's where I'd look.