I agree, tire pressure is drivers choice for ride and handling, within reason
Lower pressure allows tire's side walls to absorb some bumps, and also has a larger footprint(part of tire that is touching the ground)
i.e. lowering tire pressure down to 12psi when driving in sand for a larger footprint so less likely to sink in.
Higher pressure can make for a stiffer ride and also smaller foorprint for better MPG and less tire wear
The more tire that is touching the road the more friction there is, more friction there is means better traction but also takes more fuel to overcome that friction.
Skinny tires are better for MPG because of less weight and less friction.
Changing the weight in the vehicle can also change tire pressure, how much depends on the tire size and type.
Volume inside the tire can change as you add more weight, but the volume of air stays the same, so as weight is added the bottom of the tire can flatten out reducing the volume inside the tire, but if tire has higher side walls they also deform outward with the weight so volume doesn't change that much.
This is why tire pressure doesn't increase as much as you might think it should as weight is added.
You really don't want to drive too far with deformed sidewalls so it is always good to increase air pressure prior to loading extra weight.
If you live in an area with wide temperature changes over the seasons it isn't a bad idea to run nitrogen in the tires instead of air.
Nitrogen tire pressure doesn't vary much with temperature, like air does.
Racing cars all use nitrogen in the tires, so tire heating up doesn't change the pressure.
And you can mix air and nitrogen, no it doesn't damage anything or explode, lol.
So you can top up nitrogen filled tires with air if there is a leak.
You just start loosing the benefits of stable air pressure in all temps as the percentage of air in the tire gets higher