Generally the larger the wheel the smaller and stiffer the tire sidewall which can equate to less flex and better handling. A wider wheel will allow wider tires that offer a better contact patch.
Yep, all true and correct! I do think the manufacturers push the limit though for aesthetics, because the trade off for that sharper turn-in is increased unsprung weight which reduces suspension performance and grip. Also, having much less flex in the tyre reduces the torque feedback from the tyre when trying to find the maximum lateral grip limits of the tyre slip angle. If you want to drive around a corner using the tyres maximum grip, finding that point of approx 6-7 degrees of slip angle is much more of a knife edge with a low profile tyre. Thus why race cars which have a lot of ‘gentlemen’ drivers have such high profile tyres. For amateurs, having a higher profile tyre makes it is easier to drive fast on the limit of grip because the larger sidewall has more flex, and thus transmits that sidewall flex and twisting torque to the steering wheel to enable you to drive at that max grip limit easier. If you’re not going to push the car to this point, and you’re not chasing lap times, then having a low profile tyre is marketing BS nonsense done for aesthetics, and which gives reduced ride comfort and a higher likelihood of tyre/wheel damage due to road imperfections.
A wider tyre gives more grip up to a point. Force over area calculation means a wider contact patch also has less force pressing it into the ground. Due to tyre load sensitivity, the tyre produces optimum grip at a specific load. Wider tyres are also optimised due to the operating load they will have to cope with, especially on a car with a lot of downforce, and the tyre’s operating temperature. A too narrow tyre can overheat on track. On a road car, the width of a tyre is specced for totally different reasons. Efficiency, tramlining characteristics, steering weight, wear characreristics, etc. with the priority of maximum grip much lower down the list.