From my one hour of thinking about this problem, I think sums of finite sequences are a much more promising approach. But obviously you'd know way more about all of this than me.
This is partly why it's such a trap. It's like trisecting an angle or squaring a circle -- easy to state the problem, easy to start playing around with it. In my amateur opinion, any proof related to Collatz would be similar in complexity and non-obviousness as the proof of Fermat by Wiles, if such a proof is even possible.
"For every binary sequence of length X, the Collatz iteration ends at 1. Adding more zeros clearly does not change the result, therefor adding more ones is the only path which might disprove the conjecture." Eventually you build up to a catalog of proven sequences that all other numbers must necessarily step into at some point.