If you don't have sheeting then your wood siding is going to be structural. So if you pull it off you would have to attach sheeting to keep everything tight.
I always like blown in cellulose. Cheap and easy, albeit a bit messy! Depending on your situation, they do make vinyl siding with insulation in it, that would probably help.
But remember as a general rule heat loss happens
1. Ceiling
2. Walls
3. Windows
So take care of 1 and 2 for thousands of dollars before you attack #3 for tens of thousands of dollars.
I've known many people who have gone the blown in route and are quite happy with it; it is great for adding insulation to otherwise inaccessible spaces with minimal damage (when I help the FIL there was a 1.5-2" hole at the stop of each stub bay to be patched and painted). A down side is that if there is blocking or other large obstruction in the wall it can leave voids below them.
Since the siding is coming down for replacement, the OP will have access to the stud bays for easy installation of batting/foam board. I don't know how well blow in settle in and hold up if it was installed and subsequently the siding/exterior edge to the cavity where removed, perhaps someone can chime in, it may also depend on the way (dry/wet/dense pack) it was blown in. And of course if the insulation is being down at the same time the siding is down and the bays are exposed blown in won't really be contained.
For the windows, they could also take care of 3 with some DIY storm window inserts of with a few dollars of wood per window for the frame and whatever material fits your budget from stretch film to polycarbonate. Just because less heat is lost from windows, or that upgraded R-value windows are less of an upgrade than the other two, doesn't mean that comfort won't be greatly increase; drafty windows with a could wind can be very uncomfortable or even result in a higher thermostat setting and faster heat loss through 1&2.
To the OPs moisture point, as I understand it house wrap is more about keeping air flow down/out and providing a drainage plane for any rain that gets behind the siding. Then vapor barrier goes on the hot side of the wall, in a heating climate the inside, to prevent moisture from getting into the wall. Again as I understand it from research, it is the lack of vapor barrier that allows moisture to get in where it may condense on the newly installed insulation ... and upgraded sheathing/vinyl siding/house warp decrease air flow that would dry out the walls.
If the OP is looking for some detailed information, I suggest reading up over at
https://buildingscience.com/. They have some very good articles and other documents (though sometimes it is easier to find what you are looking for on their site with a google search than their own search).