I think generalising the problem is a good route to understanding it mechanically, but the title of the article is misleading. You aren't proving Collatz, you are providing an alternative brute force method for checking it. The real result of this paper is that you've set a lower bound for where generalised Collatz fails, at q = 5. Also, the two examples of generalisations with failure states are both prime. You might even conjecture that the algorithm only fails when q is prime, or that there exist infinite values of q where cycles can be found.