BlackRock doesn’t own the shares it holds, the customers who invest in those funds do. Buy a BlackRock ETF and watch your mail, you’ll get a letter or email about voting rights and it’s illegal for BlackRock to not provide you this notice. You can allow proxy voting, but you have to sign off on this specifically.
So BlackRock at best has a large minority ownership in some corporations, and can’t enact change Willy Nilly.