Thank you very much for the info. That's very good to know.
Having done some more testing, I think I figured out what actually caused the issue. It's not the sketch. It's such a stupid mistake. I placed my setup on a breadboard for testing, not realizing there is only one way to place the 74AHCT125 level shifter on the board without short circuiting it - across the bridge in the middle. I just put it on one side of the board. 
It'll take me a few days to actually test the proper setup (I need to desolder some stuff) but I'll definitely report back (and mark the thread as resolved) when I've got a successfully working project.