Since moving to Ubuntu 22.04 and the offical Firefox snap, downloads to /tmp do not show up in the main system. I assume because “/tmp” inside the Snap is not exposed.
There is no feedback about what locations are or are not exposed and so actually work.
I discovered my home folder worked by trial and error
Guess nobody else thinks this is an issue.
I posted a question about fine-grained directory access in late August and no response for me either. Seems like it’s just us users. 
Sorry for the lack of a response until now. That’s right, each strictly confined snap has its own private /tmp. This is by design, to prevent applications from viewing/messing with other apps’ temporary files.
This is known to break a number of use cases though, and is being tracked in bug #1790608. Feel free to add your use case there.
Definitely an annoyance for those of us who don’t care about keeping an ever-growing Downloads folder. Found a workaround by having Firefox use /var/tmp instead of /tmp, but looking forward to being able to use the latter again.