putting on a wider tire does NOT give you a larger contact patch, just one that is a different shape...
You can't be saying that ^^^^ is not true, it is
In the sterile world of theory, it is. In the real world, no one would ever keep all the other variables constant, so the old saw is wrong far more often than it's right.
So why not look at an actual case? Let's see what happens when you compare, say, a 205/55-16 RE92 on a stock wheel with a stiff sidewall, soft compound, max performance 245/40-17 on a 17x8.5 wheel--a typical WRX big meat upgrade. Then you optimize the tire pressure for maximum grip without peeling off the rim. Then measure the contact patch. Who knows--maybe it'll be bigger, maybe smaller, maybe exactly the same, but you won't find the answer from theory.