Suggested

Increase Push Level strength

I think increasing the strength of PL can add more control in race management.
It could improve overtaking outside of DRS.

A light driver who is stuck could attack a lot more because they have to push less to stay in the train.

It would improve the medium and hard compound as they can perform better at higher temperatures (it's weak now).

More emphasis on the cooling attribute.

I like that mate

But wouldn't the people in front just use exactly the same higher push to stay in front so its status quo again. Most drivers are trained to the minimum weight, it's one of the first things you do. The medium and hard tyres already overheat when it's hot weather, you can't use maximum push we have currently, still needs management to stay in the zone. Only in winter when tracks are 5 degrees could you use a higher push.

Everything is interconnected as you said, but there is a difference when the cars are not on the same strategy. In a competitive league everyone will already be pushing hard, the difference is that the faster cars (because of design, tires, fuel load) will struggle less to maintain pace because they can push less and still stay in the DRS train, giving them a chance to overtake when they decide to push.
As for the medium and hard tires, it doesn't matter if they overheat, the softs and super softs do the same but using them would be more effective, especially in warmer temperatures.

Onboard with this idea. Push levels should be key to race management. If used in the right part of the track against a lower push level there should be a noticeable affect.