How should adding chips to your stack be handled?
In accordance with whatever rules are in place at the room being played at. In the absence of stated rules, in accordance with whatever ruling is given by the dealer (or floorperson if necessary) when asked.
Part of the issue here is making any rule enforceable. People often have extra chips on their person when buying into cash games and having to monitor whether someone has added chips to their stack and additionally whether this takes them over the maximum buyin for a game is not trivial. It might be easier to enforce in an environment like a game being filmed for TV, as checks could be performed periodically to ensure the max buyin rules are not being broken, made easier by the fact they are tracking stack sizes and bet sizes anyway.
In practical terms, I've never had a problem adding chips to my stack in live cash games in Europe and the US, but I always make sure to not exceed the maximum buyin when I do this. I'm sure if I was playing a $300 max game and pulled a handful of black chips out of my pocket, someone would challenge it. But if I'm adding some red or green to a stack which still appears to be equal to or less than the table max, I think people just let it slide.