Assuming humanity exists long enough, it will expand throughout the solar system. At very least to the moon.
We can argue about how long that might take, but not about it ever happening.
That's a big assumption.
There's also the why question. What's on the moon that would make it worth all the difficulty to get there? I suppose we might have a semi-permanent research station on the moon or even Mars, but colonization? Unless we make some really unlikely discovery like unobtanium is only found on Europa and it's really super useful, we're not going to have space colonies.
The other potential industry will be moon tourism. It could become something like the new Mt Everest. Obviously only for the very rich at first.
Once it becomes a real possibility and not some crazy sci-fi project, governments may start competing, so as to not get “left behind”, even if it is not immediately profitable.
One way or another, unless we destroy ourselves, it will happen eventually. There’s a percentage of people who are just absurdly curious and adventurous, and want to go where no one has gone before, even if the cost is immense. Hell, for some, I’m sure even just desire to get away from their situation on Earth will be a big part of why they go for it. People like this will build the first settlements and bases on the moon and elsewhere in the solar system.
Edit: sorry for the constant edits. It’s a bad habit — I don’t always get my thoughts out on the first try.
And I'm not entirely sure how you turn water (H₂O) into Methane (CH₄). Granted I barely passed freshman chemistry 33 years ago so my chemistry knowledge isn't so good, but as near as I can recall, there is no process that will turn that input into that output.
And again, even if the moon is a source of water, there's not a significant need for any water mining operation at the poles to have a colony around it or even any human staffing. Putting people there on even a semi-permanent basis would likely eliminate any gains to be had from using the moon as a source of water.
Even the space tourism doesn't call for settling the moon. No one lives on top of Mount Everest either.
Also, the Lunar Gateway [2] is a key part of the plans by NASA/SLS. It's not about settling on the moon, it's about making it a stop off point to refuel or pickup supplies before going on to a further off destination like Mars. Instead of having to have all your fuel and payload when taking off from Earth, you can have a lot of your supplies and weight on the Moon. This means your trip off Earth can be cheaper. Getting out of Earths gravity and getting to escape velocity is the hard part. Getting off the moon is a lot easier.
[1] https://en.wikipedia.org/wiki/Sabatier_reaction [2] https://en.wikipedia.org/wiki/Lunar_Gateway