It looks like by running the game directly and not through itch.io's launcher, I was able to get it working. The game restart, actually, and this time instead of having my default name be like, "itch.ioplayer456543" or whatever, it was my name on my computer. But yeah, I think it's just that running it though itch.io's launcher made it get confused about something. I guess I kinda answered my own question, in that case. Thank you, though!!
I noticed that running some games through itch.io launcher creates a new user profile on the computer with a name like "itch-player-1ee23456". If loaded with itch.io launcher, OneShot seems to use this profile instead of the usual one and therefore creates the document in the folder "C:\Users\itch-player-1ee23456\Documents".
I guess that, if run directly with the .exe, the game uses the usual profile and creates the document in the expected "Documents" folder.