It really depends on how much the actual sidewalls of the tire support.
I recall a study by Avon where at the typical loads of a tire, the contact patch of wide and narrow tires were compared as well as tire pressures. For those tires, at 550 pounds the wide and narrow tires both had nearly identical contact patch areas. Above and below those figures, the wider tire did have a greater contact patch. So yes, tires are not a simple as my very simple balloon physics. The sidewall does play a role in how much rubber is on the road.
However, when they did things like double the load on the tire, the contact patch did NOT double, but only increased about 50%.
For the typical consumer, the contact patch will remain the same area, but change shape.
Of course, I also said if everything else remains the same. Seldom do folks switch to a wider tire and keep the same model of tire.
As you correctly mention, many tires with a wider width also come with a shorter sidewall, and that sidewall changes the contact patch.
But the physics is largely correct, that given the SAME tire including aspect ratio, at the same air pressure, changing ONLY the width, the contact patch will remain, MOSTLY the same area, changing only in size.
But as you said, most folks change more than just the width, and those changes do drive the area of the contact patch much more than just an area = mass/pressure relationship.
And of course, you know that bigger brakes don't necessarily mean a car will stop faster. One is typically limited by traction. What many bigger brakes will do is allow you to stop more often, as they can take more stops or stops from higher speeds before fading. It is certainly un-nerving to step on the pedal on the track (or on the street) and find that either your brakes are doing nothing, or worse, the pedal has dropped to the floor because you've boiled the fluid. But that's another story.