I have yet to see signs of "many blacks" caring about how authentic her American blackness is. I doubt most people care that she's not a descendant of American slaves, nor is Obama, who is also only half black.
The GOP would be a million times better off tying her to Biden's failures and reminding voters of all the phoniness she displayed in 2020 than pandering to racists/sexists. They have two white dudes running against a woman of color so I'm pretty sure that the racist/sexist vote is locked up and fully energized for Trump 2024. However, Trump just can't help himself.
I don't recall any cries from the left on the perceived need for Obama to "lean more into Obama's Muslim heritage."
Harris announces record-breaking $310 million July fundraising haul NPR - https://tinyurl.com/ksnbpby3
Gross. https://nypost.com/2024/08/02/us-news/chelsea-clinton-angling-for-ambassadorship-if-harris-wins/