I really like this idea, and I think you've presented a pretty amusing way to present some of the memes that have spawned from the stream. I think you should lean into it even harder in future updates (maybe make burgers and vegemite a health mechanic, kinda like the red/purple mushrooms in Mario - stuff along those lines where it makes sense).
Also, as a side note, you might consider toning down the platform spawn rate slightly. You can get fairly far in the game just by arbitrarily holding down the space bar. I suppose you could also create higher levels of difficulty to "fix" this too.