The contact area depends only on the pressure. A wide tire has the same contact area of a narrow one if the pressure is the same. The geometry of the contact area differs. It's always a rectangle but for a wide tire the longest edge is right-to-left, for the narrow one is front-to-back. This means that a narrow tire is less round than a wide one and rolls less effectively. It must be pumped to a higher pressure to be roll as well as the wider tire. So it gets a smaller contact area, transfers more vibrations to the rider and rolls the same.
In my experience a narrower tire on the same bicycle has the advantage of a lighter wheel (even if when the rim is the same, which usually is not). Two lighter wheels give quicker accelerations and a lighter bicycle.
It must be pumped to a higher pressure to be roll as well as the wider tire. So it gets a smaller contact area, transfers more vibrations to the rider and rolls the same.
I know. I added a not obvious explanation. Even now among cyclists it's common to think that higher pressure means going faster. Actually there is an optimal combination of rider and bike weight, road surface, tire, tube, rim. Pros are currently riding 25 mm tires which were anathema a few years ago.
BTW, if you optimize avoiding punctures you probably want sturdy thick heavy tires.
25 is about the sweet spot for me. I thought pros were on 19s. And the local shop only had 32s, on the regular.
Though, for me, I suspect it is largely nostalgia. 25 is about the size of the first road bike I had, and that brings a ton of good memories for how well that bike rode. I didn't realize, until then, just how different the acceleration could feel between bikes.
Oddly, I find smaller tires more comfortable. I also have better experience not popping them on debris. Expecting this is mostly confirmation bias.