Like another comment, I too have a couple of services connected to "Sign in w/ Google". Until I actually delete my Google org, those will still remain active. Since I'm not paying, I feel OK about this.

If you use a custom domain and email clients rather than webmail, moving around becomes a lot easier. Another tip is to use a service like addy.io where you can create a new alias for every service, to help protect against spam.

Have its privacy claims ever been independently audited?

https://www.fastmail.com/bug-bounty/