Wow, you excavated an ocean to cover a puddle. how this: "something which follows rules, with no independent thinking or homunculus" can be easier to prove and reason than the initial rules?
For example, what is simpler to reason out: does a chess move violate chess rules, or, there exist a method to construct an electomechanical device, that will Correctly determine whether this chess move is legal?
The reason I brought up a machine explain something as being mechanical is to clarify that it doesn't require intuition.
We can make machines that count. A trivial example: pebbles in a bucket. Neither the pebbles nor the bucket need intelligence to act as (have a correspondence with) a counter.
Completely agree about the machines. Many mathematical results become more rigorous as a result of "can be done on this kind of machine" type of proofs. However, if the goal is getting rid of intuition, machines don't help! Because no matter what the machine (bucket of pebbles or a Buchholz hydra), it takes a lot of intuition to prove, that particular machine correctly enforces intended rules. Usually more intuition, than the original problem.
Without having proof for the machine itself that machine is declared axiomatic. It is a valid way to go about things of course, but I would hesitate calling it "not requiring intuition".
For example, what is simpler to reason out: does a chess move violate chess rules, or, there exist a method to construct an electomechanical device, that will Correctly determine whether this chess move is legal?