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.

Start with the example fix in flutter/devtools#9162 and its linked discussion to understand the paused-app condition. Then audit the other service extension call sites in DevTools for calls that await responses while the connected app may be paused; done means those cases no longer hang, with appropriate coverage where existing tests support it.

Written by the indexing model from the issue text.