It could be laid out in a binding contract for the purchase of .org that the PE firm is prevented from reoffering a name renewal above the rate they previously offered it and the agreed adjustment - it just appears that isn't actually contractually laid out anywhere and is simply a PR statement.
Contract law certainly isn't something to trivially walk into, but there are ways this could have been done.
So, yes, they could absolutely start trying to shake down the biggest non-profit organizations. Those are also organizations that are highly sympathetic and have literally hundreds of millions of viewers, and thus immense political clout.