And that’s just WikiPedia.
This is a shitshow of historic proportions for the internet as a whole. It will be remembered as a landmark in the ever metastasising cancer of monetisation that is suffocating the World Wide Web from within.
Wikipedia sized organizations would easily absorb the relatively trivial cost, so, they wouldn't switch. Their demand is relatively inelastic to price.
The people switching are going to be the far more numerous smaller operations, who are elastic to price.
And, of course, anyone signing up for a new domain is highly elastic.
Because there are lots of people who are highly elastic, the overall demand is highly elastic.
Maybe Wikipedia gets it's registration price bumped up to 3 million because the PE firm thinks the org can absorb the cost.
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.
Contract law certainly isn't something to trivially walk into, but there are ways this could have been done.
Competition with all the other registries?
For the most cost sensitive domain owners, they typically have a presence on social media and contact with users via email. And, of course, they're usually discovered by search engine rather than domain. So the switching costs are quite low and the demand extremely elastic.