chore(Turborepo): Consolidate Subscriber::watch and setup task (#7714)
### Description
- Remove the async task in Subscriber::new, move functionality to an
init_watch function
- Cleanup based on no longer needing the one-off async task, including
removing unnecessary Arcs
### Testing Instructions
Existing test suite updated with new arguments
Closes TURBO-2626
Co-authored-by: Greg Soltis <Greg Soltis>