Are there passenger flights with stops in the middle?
These &quot;direct&quot; flights are actually not unusual, at least in the US. I was surprised by this discovery based on the routes I personally tend to fly, but fortunately we have the Bureau of Transportation Statistics to provide objective information. Using the July 2019 data for all major carriers, I identified direct flights by counting multiples of the same flight number on the same day with the same aircraft, as opposed to single-segment &quot;nonstop&quot; flights. Here's what I found:

Some interesting notes:

Over a third of all flights are direct.
The practice is not limited to low-cost carriers. In fact, the two largest traditional carriers (American and Delta) operate around half of their flights direct and together operate almost two-thirds of all direct flights.
Nevertheless, Southwest uses more direct flights for its size than any other airline, nearly rounding out the other third.
On the other hand, JetBlue, which is commonly considered a low-cost carrier, barely operates any direct flights. Other low-cost carriers use anything from zero (Allegiant) to about a quarter (Spirit, Frontier) direct flights.
The other traditional carriers use direct flights sparingly. Hawaiian flies only nonstop routes, which makes sense.

I may have missed a few direct segments that departed past 12am on the day after the previous segment, but I don't think there's enough of those to substantially alter the results. In any case it would just increase the number of direct legs.
It'd be interesting to run some more analyses to see historical/annual trends in the practice, but that would require someone with more patience and/or computing power than I have. These data sets are huge and can only be downloaded month by month. I chose July 2019 because that was about the peak of commercial passenger aviation.
There's also some fun aviation memorabilia associated with these &quot;direct&quot; flights:
(source)
On airlines with no assigned seats, plastic cards like this used to be provided so that even the passengers continuing on could deplane if time permitted. Somewhere I have a couple old ones from Alaska Airlines. Nowadays I believe the airsickness bag doubles for this purpose in some cases. I suppose if you really didn't want someone to take your seat, you could leave a full one behind.
Note: A previous version of this answer had a much lower count of direct flights. Turns out Excel just wasn't done calculating yet.