Meanwhile, I think there should be a tea line to each seat with a spigot and a mug supplied when you board, but then I have a teapot on my desk at work for efficiency, which I refill several times each day.

AA crews are actually better at this than BA for me, usually once they notice that my tea consumption on a daytime flight is mugs/hour (especially when working) they refill me quite diligently. While in theory BA will give you tea when you like, in practice it's definitely only served on schedule at the end of the meal.