The key is to do lots of problems. Toward this, I highly recommend "The Art of Problem Solving" which is packed full of neat insights and interesting problems from math competitions (at an "advanced high school" level).

https://www.amazon.com/Art-Problem-Solving-Vol-Basics/dp/097...

That's super weird. In eighth grade (2013-ish, not long ago) we had an entire course on geometry that was all about proving theorems and turning some given information into "the shape satisfies the definition of a parallelogram because..." The tests were exactly what you described you wanted: hour long journeys where you applied the theorems and postulates we learned up to that point to prove something about a shape. I struggled a lot but ultimately I believe I am better at thinking because of it. If you want some books:

The textbook we used: https://www.amazon.com/Geometry-Ray-C-Jurgensen/dp/039597727...

The Art of Problem Solving: https://www.amazon.com/Art-Problem-Solving-Vol-Basics/dp/097...

AoPS is pretty good but more geared toward math competitions. The reading is ok but the problems take some serious thought and if you can get your kid to do one a day s/he'll probably be better off for it.