Hi @nabulon,
Thanks for your help with this. I've figured out what was happening, and it turns out this wasn't an issue with the Celery workers. Both of my Weblate instances were behaving as designed.
The problem was that the entries from my uploaded TMX file were being set to "pending" status. According to the documentation (https://docs.weblate.org/en/latest/admin/memory.html), pending entries receive a quality penalty.
This penalty was significant enough that even my exact matches were rated too low to appear in automatic suggestions or be used for automatic translation.
I confirmed this when I manually translated one of the strings—after that, the automatic suggestion from the TM finally appeared for identical strings.
Thanks again for pointing me in the right direction!