Lawn damage after treatment - Roll turf approximately 2 years old

That's not a dust desert yet, depending on your standards, you can just leave it as it is. I always do that, it turns green again during wetter periods. And I even park my G-Class there ;).

Instead of the G-Class, I prefer to park my grandchildren on the 2-week-old sod ;)
Watered twice daily, very well rooted/merged, though slightly uneven in height, but lush green. Probably mowing for the first time tomorrow...

Nothing needs to be overseeded on a putting green.

Yes, it does. I have removed Poa Annua and, of course, there are open spots left.

If the lawn is mowed too short and too often, it will burn. No matter which lawn. So less is more.
Also, the lawn needs to be watered more often on hot days. The shown [Rollrasen] has a water problem, not a fertilizer problem.
It also depends a bit on the substrate. If it has been heavily compacted, no moisture is retained there and the lawn will burn as well.
So water properly and it will look properly green again in 3-4 days.