broadcast: Update source instead of remove for no context

Refactor streaming request handling to enable resumption after unicast
suspension. Instead of removing and re-adding sources, modify existing
requests when no context is available. This simplifies the logic and
improves stream resumption after unicast interruptions.

Bug: 395713415
Flag: EXEMPT day-to-day bugfix covered with unit tests
Test: atest BassClientServiceTest
Change-Id: I87f0bdb1c77b7ddb3aaa09f24e214c75884765b7