Yes, you would need an actual email provider to do this typically.
The server delegated to handling incoming mail for a domain is defined in the MX records, but that server doesn’t have to be the one sending the mail out. If you sign up for another mail provider and add their DKIM record and adjust your SPF to include both Cloudflare and their SPF record, you should be able to send emails out using their service but incoming will route via Email Routing.
one more question
do all inbound email addresses created via email router in Cloudflare require to have corresponding address on 3rd party mail service provider?
or that is only requires for sending emails via domain address?
i.e. if [email protected] is created only to receive emails, does it require to also be setup on mail service provider?