The odds are actually pretty high that you will eventually "meet" somebody, as you can see. The odds of you being on the same planet at the same time and in the same place are where the odds drop like a rock unless you put in the effort and materials.
This is actually a version of the Birthday Paradox. With enough people involved the odds of two people having seen the same planet goes up VERY quickly.
Answering my own question here, it turns out to have been, if not a glitch, at least a bad interaction due to a corner case.
What happened was that, after I had placed the Agricultural Station in my base, I was editing my base structure from outside. I briefly misplaced a structure so that it overwrote the Agricultural Station, but since I was outside, I didn't notice.
Apparently the result of this—since I'd already activated the Farmer part of the mission and it had told me to go find it—was that the Farmer showed up in the system, but jumped away from the place—station or planet—I was on so I couldn't reach it.
Once I went back inside my base (having given up, for now, on the Farmer), I noticed the Agricultural Station was missing.¹ I reconstructed it, went back to the station, and:
—
¹ This may be useful for some: if your inventory is filled in the process of deleting an element (because you have space in your inventory for the main item, but not subitems like glass for windows or you fill a stack when there's no slot in your exosuit to start another), you can build the Base Salvage Capsule, which is a large machine like a Large Refiner. It will contain any such items. (It can hold a huge amount, but allows only a one-way transfer to your exosuit.)
Best Answer
At least on PC (in the Beyond update), the 5 missions per day is a game limitation that has nothing to do with the size of your freighter.