A quick little update, as I had time to play around with stuff.

Leaving and rejoining the queue had the same exact behavior. I tried rejoining three times, waited 40m, and 25m both times.

After about 45 seconds, the UI would go 0/2 DPS from 1/2 every time. No idea if this is linked.

Now, when I queued all available dungeons manually (that the leveling roulette would have selected automatically) not only did I not get the 0/2 bug, but I queued in within 5 minutes.

Roulette is definitely broken and that really sucks for the 20's grind before you can start using FATEs efficiently.