You asked "Is the label so important to claim?"
Ok, that. Sorry. What you are referring to is more clear now. The alignments are important. Claiming the labels are less so. Like does my argument become clearer to you if we label Good, Weal, and Evil, as Woe? I personally think that only makes things less clear, but my point is that Good isn't obviously correct. I mean even in the real world it's not at all clear why in a universe with no Justice particles one ought to be Just, or why in a universe with no Charity particles one ought to be charitable. It's not clear why one ought to do anything. But even in a universe with Good and Evil particles (as it were) that you could detect, it's not at all clear that choosing what is Good would be right. Why ought you sacrifice your own interests for the good of others anyway? Why does it profit to care for the weak instead of cull them? Why should you value love over power in the first place? Those answers aren't addressed just because we have Good and Evil objectively defined.
I feel so much of the argument comes down to "my opinion about what is Good must be the prevailing one at the table and it's objectional to me if it doesn't". Well, I got to tell you, my opinion about what is good is almost never the prevailing one, in real life or the game, so that's not really an unusual experience for me. I have to deal with that all the time in situations much more important than merely a game. My advice is lean into it. If the DM's universe judges you a monster, well rail against the universe. Demand the universe be accountable to you. Cry out to the gods and judge them in return. And if the problem is the DM isn't God, but is the DM is the Devil and is not interested in running a fun table, well that problem can't be solved by anything or any rules or lack of them. Nothing can protect you from a bad GM.
You said "Yes, in D&D there is an objective answer to what is meant by "Good". I'm not sure how to take that in a way that isn't equivalent to it somehow. (Even if only the god/DM who does play dice with the universe).
I feel I keep explaining this over and over. In the Great Wheel cosmology the gods sitting in the Seven Heavens don't have a privileged place at the table. Theres is just one take of many and they aren't even omniscient or omnipotent. How do you know that they are more right and correct about how life should be lived than powers in the Abyss? How do you know one ought to be Good? If I tell you how Good sees the trolley problem, how do you know there answer is better than how Law or Chaos sees the trolley problem? For that matter, how do you know that a neutral evil fiend's answer that of course you should choose whatever results in the most harm is wrong?