Pardes, if you're not mountain biking - I think flats happen less often with mountain bike tires with larger tires.
That has not been my experience with the fat knobby tires on our baby trailer ... it gets flats all the time! But my theory is that it is because it comes closer to the edge of the trail, where thorns are more likely to have gotten onto the pavement, and because if I have to pull off the side (as I do, often, to hand somebody a dropped sippy cup) the trailer wheels go a bit further off onto the shoulder.

But I have had one flat on my road bike ever, riding mostly on the same trail where we use the trailer, so I don't really have a sense that the fat knobbies flat less. Pretty much the opposite.

We're going to put liners in those tires. In fact, I need to nag my husband to do that.