Formula One ( F1 )

I also don't understand how running longer on a set of tyres helps with aero set-up/ balance improvements.

Maybe because the aero set-up/balance improves as the tyres degrade?

We do know that there's an optimum period for the tyres to perform well in. Maybe Hamilton went through this period quicker whilst driving harder than Button?

Gridds will know better than any of us.

It's basically about understanding performance changes as the tyres wear. As well as grip levels the aerodynamics also change. As the tyre wears it will reduce in diameter slighlty which will change things like ride height and the gaps to the bodywork and floor at the rear etc. There's also the influence of any lips that are formed on tyres as the rubber rolls off the sides of the tyre. This will affect the vortices coming off the tyres. So these seemingly small things have a significant influence on the aerodynamics and hence balance (rear/forward downforce levels).

Understanding these influences are why they would maybe want to run on tyres in these conditions as it's something you can't test in the tunnel but would be benficial to know in the event that you would need to run on worn tyres from a strategy perspective. Or how to minimise any adverse effects.

Added to that it's about how to work the tyres properly, i.e. getting them to and keeping them in their optimum temperature performance window and how to maintain that as the tyres wear. As well as mechanical set up, aerodynamics plays a significant role in how the tyres are worked.