Relying on an oppressive walled-garden corp like Cloudflare who sees all traffic is an instant show-stopper for me. If CF is necessary to keep it gratis, then we have to wonder how Cloudfare is profitting. As you say:
“That, and Cloudflare is itself somewhat of a tracking and analytics company; so you're never going to be really anonymous using this setup.”
I only read that far. But I have to say getting your own domain should be sufficient because some domain name services include gratis email forwarding. I don’t recall having a limit on the number of aliases but my memory could be off on that.