Why did a lot of websites all of a sudden switch to hCaptcha?

Works better at stopping bots, costs less than reCAPTCHA Enterprise since it doesn't need to store your browsing history for the last decade.

(disclosure: work there)

Google started charging for captchas.

They haven't, you're probably just running into cloudflare captchas.

I see it everywhere, even Cloudflare switched. Was it a privacy thing?

hCaptcha isn’t run by Google. That’s reason enough for me.