[extension-browser] Refactor devtools panel to not call sendMessage directly

[antross]

I've been debating refactoring this a bit to have a single utils file with methods to call for each action that sends messages (and only have that file actually call sendMessage).

Originally posted by @antross in #2140

This should help the code that triggers messages read better and provide a clear choke point if we need to change how messages are sent (e.g. if we move some analysis to a worker in the future)