Is Duty Finder's "Join Party in Progress" feature broken?

[Iscah]

But they also didn't get anyone else and had to finish the dungeon with 3 people only. We had finished the mobs after the first boss when my internet issues started.

That just means that nobody else queued during that time period with "join in progress" active and the ability to join that high level dungeon, or perhaps if they did, they got scared off by the high running time so far. It's also possible that MSQ players don't want to join a new story dungeon halfway through, and it is quite likely that people are playing with trust NPCs at this point.

In any case – regardless of your specific niche circumstances – most parties who have just kicked someone will have done so because they don't want that person in their party, and it would make no sense for the system to put the same person back in just so the party has to kick them again.