Also some cars I've owned owner's manuals have advised that even non-directional tyres should be regarded as directional once they've been on the car a while.
I read that some years ago as well.
I think it relates to the tyres getting a 'Wear' pattern, as the wear is not usually 100% even across the tyre. (This happens even if the tyres are inflated correctly etc)
I have noticed this on a car when I was doing a regular journey of hundreds of miles, twice each week with lots of Roundabouts\winding roads and the some of the tyres were wearing more on the edges.
(Yes the tyres were inflated correctly and the toe-in/out etc was correct. Company Car, so everything was serviced and checked to the letter.)
This also relates to Tyre rotation which I have not noticed in car manuals for years.
It could also be the Tyre Manufacturers NOT encouraging you to make your tyres 'last longer via rotation' to increase sales/profit.
As I trust my life to Tyres/Brakes I tend not to take risks and do not run them down to the last mm of safe tread/wear, even if the law allows me to.
I also, personally, would not run on Tyres with a noticeable asymmetric wear pattern and would not rotate them to try to even them up.
(Until you manage to get the Tyre wear even across the tread, you are actually running on degraded grip etc. Also by time you have 'evened out' the wear you may be down to a tread depth that is giving you degraded grip anyway.)
I have noticed that Tyres give you their best performance during the 1st (1/3 to 1/2) of the Tread from new, after that you are adapting your driving style to the reducing performance of the tyres.
That is why you notice such a change when new tyres are fitted, as you have been compensating for your Tyres as they wear !!