Also it’s easier to be confident that your distilled water is still mostly pure as the manufacturer will be regularly checking their output. Unless you’re running regular testing on your reverse osmosis equipment, and cleaning it and the downstream pipes, then how you be certain about the purity of your home made water?
If you are doing all that work, then home made purified water from a reverse osmosis setup is probably as good as store bought distilled water.
What I REALLY would not use is an old Britta that's been accumulating goo for the last year. Fungus is bad, hard to kill, and will eat your brain. Do you have sourdough, kombucha, or any other weird smells in your fridge? Keep it outta there!
You nailed it. Misuse can be fatal due to proximity to the brain.
(and of course, as others have said, bacteria, amoeba, disinfecting chemicals, minerals from the house's pipes... did I mention that I'm not paranoid? :) :) )
But YMMV -- if you're happy with reverse osmosis, go for it.
edit: and yes... look at the news every now and again where you hear of some kids swimming in a lake and dying from an amoeba or something else that jumped the brain barrier.