Well, I disagree that it's completely unapproachable. And I take a slight bit of offense that you think my comments are disingenuous. You don't sit down to "solve" an open problem. You sit down to learn things about it. And your claim that you can't learn anything useful or discover it's intractability is just flat wrong.
Collatz has two parts, one that there are no cycles, and two that there are no divergent trajectories.
For example, you can trivially prove that there are no cycles of length 2, or even 3, or 4. It's not hard, at all. It's not unapproachable. It's easy. For a slightly more difficult problem, you can try to generalize your method of proof to show there are no cycles less than some number N, given some criteria. I'll leave that as an exercise to the reader.
I've developed my own proof about the smallest possible cycle. It was actually pretty good. Once I decided it was legit, after alot of work and refinement. I read a bunch of papers on the issue from actual mathematicians. Mine wasn't as good as theirs, but still not bad.
Or, you can categorize numbers by how often, after undergoing the 3x+1 step, they divide by 2. For example, the number 3 only gets divided by 2 once (3->10->5). You can very quickly find the patterns to show all the numbers that divide by 2 once, twice, and so on.
You can go farther with this pattern idea, as well... by finding the sequences of numbers that, for example, divide once on the first iteration of the Collatz function and then divide twice.
Using this idea, you can prove, for example, that there are no divergent sequences that only divide by 2 once at each step.
I have a few dozen pages in a notebook with a whole bunch of interesting but likely useless facts about Collatz. So please don't tell me it's "unapproachable" and nothing can be done except stare at it dumbfounded.