If I could show you an example of mathematicians running ongoing computer simulations in order to test theories (well. Test conjectures for progressively higher values), would that demonstrate otherwise to you?

This happens, but the conclusion is different. No matter how many cases of an infinite-case conjecture I test, it's not going to be accepted as proof or even particularly valid evidence that the conjecture is true. The point of doing this is more to check if there are any easy counter-examples, or to figure out what's going on in greater detail, but then you go back and prove it.