Yes, it’s worth it. I own mine for just the reason you give. You can take it to any other provider. And there’s no danger of the email provider deciding to close your account or cutting you off unexpectedly. Imagine losing your email access. At least with your own domain, you can switch it that same day to someone else.
Unsure about whois lookup privacy. My registrar hides my details as an option. Anyone looking up the domain just sees them as the contact for the domain.
This better be an April Fools joke