[Death during Eidolon hunt causes rewards from all subsequent hunts to be lost]

In this case I was definitely missing rewards, I had no sentient cores before starting the two tridolon hunts and none after.

I think this is narrowing down the bug though, I was host and your video looks very much how I remember it.

[Death during Eidolon hunt causes rewards from all subsequent hunts to be lost]

If I have the patience this weekend I might try to get a video of the bug being reproduced.

[Death during Eidolon hunt causes rewards from all subsequent hunts to be lost]

This has happened twice to me now, first time during a Teralyst hunt and second time during Tridolon. The steps to reproduce seem to be as follows:
1. Start an eidolon hunt with a squad.
2. Die [repeatedly and use up all your revives]*, then allow yourself to bleed out.
3. Your squad completes the hunt and returns to the Cetus gate.
4. You respawn inside the gate as Cetus loads.
5. The mission failed sound plays twice and the post mission screen appears to show once, disappear and then show again.
6. You will have no rewards from the hunt.
7. Start a new eidolon hunt immediately.
8. Complete the eidolon hunt without dying.
9. You will have no rewards from the hunt.

*I'm not sure if using up all self revives is necessary to reproduce but it has been the case both times I experienced the bug

 

This behaviour at step 6. seems strange already but it could be intended behaviour to lose all rewards if you die during an Eidolon hunt.

Losing the rewards from a separate unrelated hunt to the one you die in is definitely not intended.