If you switch to , we’d have to play a bit with Google Sites settings HTTPS/SSL and SSL at Cloudflare to get your Website working.
I’ve done it here (disabled HTTPS on Google Sites, proxied both DNS records and with Flexible SSL with Configuration Rules added 2nd A www and created a Redirection Rule from non-www to www or vice-versa):
Using above, you’d be ready to have Google Sites over proxied both non-www and www, including redirection from www to non-www (or vice-versa) and once you need the Redirect Rule to YouTube would work with single click to enable it.