(Un-ciphered spoilers after this)
Thanks for your reply. I got to step 4 on my own before making my first comment, and my intuition told me the exact thing I need to prove after that.
I did consider one form of parity, which is the numerical parity of horizontal/vertical lines. I thought the total number of either horizontal or vertical lines must be even because you must draw 2 lines per move, which could prove only one possibility works, but that's wrong because you can go from 2 to 0 by drawing "1 line", so I couldn't prove step 5 as easily as you have described.
But now that you mention it, I notice that there's also "positional parity", which allows me to color every (vertical) line with black or white, then notice you must draw 1 line in both black and white per move considering the relative positions of the astronauts. Drawing one line can cause the total number of lines go +1/-2 as mentioned before, and after all moves, the difference between black and white line numbers will always be a multiple of 3, finally proving that a difference of 1 in that other two possibilities can't ever happen.
I do feel satisfied with my proof no matter if it is the intended thought process or not, maybe there's an easier way to tell that? But if it is, I have to say you are evil LOL