The reason why you see the second sentence is that warmongering diplo modifiers are usually much, much bigger than all others.
I think that "second sentence" is the focus of the complaints. And I think you are underrating it by saying warmonger modifier is "much, much bigger" -- reality is that the instant you take or raze a city it makes all the other numbers completely irrelevant, might as well not even list them.
The builder crowd likes it since it matches their philosophy and doesn't really affect their games, but imagine if you built one campus then every other civ instantly hated you for the rest of the game? That is the path of the warmonger in Civ VI, when what warmongers would rather see is a more sophisticated system that acknowledges that sometimes a bad-acting Civ deserves what it gets if it declares war on another Civ, or sometimes some Civs will be happy when you burn their sworn enemy's cities to the ground.
In past Civ games you could roleplay shifting alliances in strategic wars, now if you choose to war you have basically chosen always war against every other civ, there's no in between and as the OP says, no realpolitik to war in Civ VI. Indeed, to now see Gengis, Shaka or Monty denounce the player for warmonging is a blaspheme kick in the pants to an old-school Civ warmonger, and takes away a lot of the fun!
Anyway, to the OP, this philosophy of "if you take a city you are now the aggressor no matter what" is simply what the game has chosen and celebrated by the builder crowd, which I think is unfortunate since we all should want every player type to have more fun, not less. Personally I think the real reason why it doesn't work in a more sophisticated way is the AI cannot competently conduct warfare, so domination is too easy, or they don't trust themselves to implement such a system that wouldn't be open to exploitation. So instead of fixing the actual problems, they add artificial barriers to warmongering. Whatever the case, the thing that warmongers want most -- an actual strategic challenge -- is no longer in Civ. Sure cynics will say it never was, but at least maybe they faked it better in past Civ iterations.