After spending the entire afternoon doing various in-depth tests, my conclusion is that sometimes it works and sometimes it doesn't. I don't know who's to blame, the simulator or the add-on, but I haven't found a logical explanation.
I used the same settings in both the simulator and FS Traffic. Sometimes traffic is displayed and sometimes it isn't, both on the ground and in flight.
I've tried larger airports (where it's impossible not to have ground traffic), doing shorter flights and slightly longer ones. I've changed the time of day, changed options in FS Traffic, like Time Before Departure Value, deleted traffic, and then regenerated it, but there doesn't seem to be a pattern.
When it works, everything works fine, and when it doesn't work from the start, no matter what I do, it doesn't work.
The truth is it's a bit frustrating and the program's behavior is quite random.