I noticed that when I'm running an app with an active Network Extension in macOS Monterey (21A5268h), changing the Private Relay state to on or off disrupts internet connectivity. The WiFi menubar icon eventually turns into an exclamation-mark.
The only workaround I found is to turn the filter off or to delete it. If the filter is turned on again, Internet connectivity is lost. I've tested this with a pass-through NEFilterDataProvider-based NetworkExtension filter app, but also see this behavior with other network filtering apps such as TripMode which is on the Mac App Store.
Is this a bug? I've filed a report (FB9283392) and attached a sample project to it.