Cloudflare has Email Routing Cloudflare Email Routing to forward emails to external mailboxes, but not reply, and forwarding is so-so, I wouldn’t do it on any critical emails unless you have full control over the destination and can ensure the forwarded emails won’t get spammed/rejected (ex. not gmail).
Otherwise for a full inbox/sending and receiving you’ll have to stick to an external service. Fastmail, Google Workspace, Microsoft 365, etc.