agent_servers: Use built-in interrupt handling for Claude sessions (#35154)

We no longer have to stop and restart the entire process. 
I left in the Start/Resume mode handling since we will likely need to
handle restarting Claude in other situations.

Release Notes:

- N/A
a5b7cfd128Ben Brandt committed on 7/27/2025, 1:50:04 PM· committed by GitHubparent89e88c2
3 files changedLine totals unavailable