Fix spectator not immediately showing when a spectated user fails

[peppy]

Supersedes and closes #25550.

• Depends on Fix spectator mode not showing when a spectated user quits correctly #25565.

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CleanShot.2023-11-24.at.06.07.20.mp4

[bdach]

The source here looks okay, but:

• I have had this fail to trigger failure for me on the very first try of this and I am unable to reproduce it which is unnerving.
• The retry button on the spectating player's side does things it shouldn't:

2023-11-27.11-54-40.mp4

Also it looks that if the spectated player quits at the retry overlay, the spectating player will not be sent back to the spectator screen.

[bdach]

I have had this fail to trigger failure for me on the very first try of this and I am unable to reproduce it which is unnerving.

It has not failed since in several tries, so I will pretend I did not see that...

The retry button on the spectating player's side does things it shouldn't:

Fixed by cherry-picking a4be28a out of previous iteration.

Also it looks that if the spectated player quits at the retry overlay, the spectating player will not be sent back to the spectator screen.

As discussed IRL last week (and then promptly forgotten by me since I have the memory of a goldfish), this is not simple to make happen because the spectator server has no way of knowing about the quit (the player has already "ended playing" by then, by way of failing).