This kind of thing used to be used extensively in railway signal boxes, where you want to enforce safety constraints like "signal 1 clear => points (switches) 20 set to straight ahead AND signal 2 at danger".

The UK version of this is called "tappet interlocking".

What I find odd about this is having to push the output rod and check where it ends up.

I'm not sure how this would work in a clocked/latched system, or whether that's even possible? Does it need a reset at the end of a cycle, or can it only live in a async clockless circuit?

[edit] OK I searched and found that the Zuse Z1 used this system too! And had a clocking system. This page describes the same problem and has details of alternative designs :

https://anthony-zhang.me/blog/rod-logic/

Another objection to the first OR gate: it's actually an AND gate, isn't it?

You have to imagine that the T-shape is pushed by only one of the blue rods and loses contact to the other blue rod.

Aha!