Taylor Swift does tend to make everything about her, or others do it for her. I doubt she's written the post herself (I haven't seen it).
As others have said, it's just a very annoying aspect of modern life. I remember in the late 70s when very young, if someone passed away, my dad would groan "oh nooo" and call my mum in to watch the report. It would then be in the paper the next day, and then life would move on. No one sought attention, and the means to do so didn't exist.
Today we had Richard Madeley very close to bursting into tears on GMB! People also like to act like they're the biggest fan of someone who's passed away. I remember when Michael Jackson died, a friend of a friend on social media posted that she was "sick" with grief. It later turned out she didn't know any of his songs (apart from I guess the obvious ones). Same when David Bowie passed.
It is just sadly modern life, when people love to make everything about them, most people love attention, engagement and "likes". People need validation.
Dolly's passing is a huge loss, I am a massive fan, but I haven't said a word to anyone except my partner. But others are going to. It's just the way it is.