I want to stress that I actually don't give a shit (I am not that into tea, I just drink it sometimes, it's not a big deal to me), but I am curious as to why.

Unrelated to the tea, but cold water is somewhat safer to drink than water coming from your boiler, that's why.

IDK, they said "a terrible way to make tea", I'm assuming there's some sorta tea lore that I'm unaware of

There are tea drinkers that believe very strongly in using water that isn't "stale" (eg, leftover in the kettle from the last time). Presumably water from the hot water tap would be stale for similar reasons.

I think the theory is that "stale" water lacks sufficient oxygen for tea brewing to fully brew or something.

It's not O2: if you let water sit out, various things (including CO2) dissolve in it, making it slightly acidic and taste stale.

I'm not a big tea drinker, but I am definitely not a super fan of water that's been sitting out a day. There's not been time for stuff to grow in it yet, but it still tastes gross.

That makes sense.

Fill a cup with cool water and take a sip before you go to sleep. Tastes great, right? Taste it eight hours later when you get up in the morning. It takes on an acrid taste.

It's not hot enough. Black tea is usually brewed with boiling hot water around 95 - 100 °C. Green tea is usually brewed a bit lower than that at around 80 °C. But the domestic hot water supply is typically around 50 °C, to prevent burns. It'll be very weak, and probably taste a bit off. Same with coffee, or anything else that requires boiling hot water.

I don't drink tea made from hot water. I put hot water into a pot and heat it up further.