Ensure we do not call service extensions when the app may be paused

If we attempt to call a service extension while the connected app is paused and we are awaiting the response, DevTools will hang. https://github.com/flutter/devtools/pull/9162 is an example fix for one of these issues. We should audit other service extension calls in DevTools to make sure we aren't doing this elsewhere.

See comment here for context: https://github.com/flutter/devtools/pull/9162#discussion_r2070633192.