The story said he cares about his crew members. It didn't say to disregard anything that was important to them. Thatch was a lost cause. Fighting Teach or not wouldn't bring him back, but Newgate wouldn't budge over a friend Ace made and knowing they come from Oden's home. Remember that Newgate had a bunch of allied crews who were all indebted to him. Meaning he helps those he considers friends.
Then I guess you don't care for your argument.
There's nothing that states the whole world knows this. Newgate could've just assumed Oden was dead because Kaido was still in power.
The story said he cares about their safety, Oden and family were dead.
It also never said he cares about their feelings and what is important to them.
Based on?
Yep, might have saved those allied people, not protected what is important to them.
No, i do not care whether it is considered a weak trait of WB.
If you want to spin around words, this is meaningless.
The world knowing is irrelevant, the WB Pirates somehow got the info that Oden is dead and Wa No is in that state.
With that info, it being completely disconnected to the demise of Kozuki, highly unlikely.
Anyway, you are going into hypotheticals which is equally meaningless.