[ty] Make sure to always respond to client requests (#19277)
## Summary
This PR fixes a bug that didn't return a response to the client if the
document snapshotting failed.
This is resolved by making sure that the server always creates the
document snapshot and embed the any failures inside the snapshot.
Closes: astral-sh/ty#798
## Test Plan
Using the test case as described in the linked issue:
https://github.com/user-attachments/assets/f32833f8-03e5-4641-8c7f-2a536fe2e270
