I do this as a matter of routine. The benefits are crystal clear. On a front heavy car in the UK the front left tyre tends to take a battering, and the front in general is doing more work since weight transfer under braking places most of the load on them. Rotating tyres tends to make the entire set wear at an even rate.
Makers, including European makers, used to put detailed directions for rotation in the owners manuals in the servicing section. I’ve seen it and followed it on everything from a mk3 cortina, a ‘79 Scirocco, and an E28 5 series, up to more recent metal such as my old Mazda 6 MPS and my last Audi. The 4WD system on those cars were particularly fussy about tyres, and only worked properly when all the tyres were at an approximately even state of wear.
Other cars I’ve owned had different sized tyres front and rear so rotation wasn’t possible. Directional tyres make only front to rear swapping possible without remounting tyres to rims, but it’s still beneficial.
The advantage of course is that for the life of the tyres the even wear means they all offer consistent and even levels of grip, slip, and general performance, maintaining the balance of the car when accelerating, cornering and braking. This is an area of chassis dynamics not often considered outside of racetracks, but it is important. Put a pair of very expensive, very sticky go faster tyres on one end of your car, and a set of budget or part works on the other and your performance will be limited to the capability of the lesser tyres. The car will be unbalanced, moving more readily to over or understeer at a reduced limit determined by the capabilities of the lesser tyres, and not only is the car less stable, but you’ll never get the benefit of that extra investment. Think of it as trying to run the 100m sprint with a the latest in featherweight Usain Bolt spec running shoes on one foot and a flip flop on the other.
Proper rotation also means all four tyres are then due for replacement at the same time, so your new tyres are matched, and you are free to opt for a completely different tyre type if someone else ever makes a better tyre than the Michelin PS4S. You get balance, and you get to enjoy the benefits of having 4, perfectly matched brand new tyres on the car again.