You're wrong. You're arguing from surface similarity rather than detailed internal workings, which is a big no-no in maths. I could have spent about five paragraphs breaking down your argument point by point, but you've made my life easy with that final line:

therefore it is able to evaluate everything needed for Godel's proof, therefore it must have not only one but an infinite number of cases for which it hangs.

This is not the case, there is a simple counter-example. Unary code, in which 0 is 0, 1 is 10, 2 is 110, 3 is 1110, 15 is 1111111111111110, hangs on the string 111111.... and no other. The fact your argument led to this conclusion is a demonstration of how completely wrong it is.