No, it‘s not just that. I would argue that we need to be clear whether we compare tires at the same pressures or at equivalent pressures (i. e. choosing pressures so that the surface tension in both tires is the same). I would argue you should compare tires at equivalent pressures, because the casing tension determines the size of the contact patch. Wider tires will have a lower equivalent pressure compared to a narrower tire, which improves compliance and therefore grip at the same time. Running a wider tire at the same pressure will decrease the size of the contact patch, but not necessarily decrease the friction.
There is also way more to rolling resistance than your static considerations, precisely, because tires are in motion, not at rest. For example, wider tires run at equivalent pressures deform more, which means they have more friction. But this is more than compensated for by impedance losses, which are a dynamical phenomenon. That is why there is no the optimal tire pressure, the point where you minimize the sum of casing losses and impedance losses depends crucially on the surface you roll over. Plus, the rougher the surface, the more of a premium I would put on compliance, not just to maximize comfort, but also grip and safety.
There are also other factors to consider when e. g. include cornering, because then tire shape, tread pattern, internal rim width and other things become important. Or if we just look at compliance, casing material, thickness, rubber compound, etc. all become factors, too.
That‘s the wrong car tire analogy. The proper analogy would be changing rim size: if you go for smaller rims with taller tires vs. larger rims with shorter tires (keeping tire diameter constant in the process), you will have more compliance and comfort on smaller rims with taller tires.