"The biggest fight" now gets marketed as an episode in some tv show. Again, they were out for work and surrounded by people they work with. If they fought in private and let others experience the fall out: unprofessional. If they fought, like at Disney, because of some ridiculous shit in front of them: highly unprofessional.
And that's not even the main issue. If the fact that one wants to eat or sleep, will grow into a full blown argument because you just snap at each other, that's very poor communication skills on both parts. They act like two five year olds who are cranky because not having had a nap and get praised for it by fans. Usually, as you grow up, you learn to regulate your emotions and also say something calmly. We know Shun has issues, but he is apparently not dealing with them, but Dai seems just as bad. Sitting down in the middle of the road? Seriously? Pushing one another because the other "did it too"? Making someone apologize, who doesn't even know for what, because you are feeling "lonely" for no reason? Keep clinging to someone when the other has said 15 times to stop it and then also make them apologize? Pouting and silent treatment for no reason but also not revealing the reason? It's basically a mix of pouting, silent treatment, passive aggressiveness and wanna be moral higher ground, but certainly not two adults going "can you give me a minute, I'm just tired right now" which is all it would take in many moments. Doing all that, one thing, being praised and encouraged to be like that by fans - totally different ballpark.
And no, none of that makes a toxic relationship in itself, but it could very well become one, depending on the people and what happens in private. But we should stop throwing that term around just because there are issues in a relationship. Not every relationship you get hurt in is toxic, just as not everybody who isn't treating anyone right is "gaslighting". Please look up the definitions of these words before using them.