I know I can do this with a reverse proxy like Cloudflare, but I don't need a reverse proxy necessarily.
It would be the DNS service that Cloudflare provides that would allow you to do this, not the "reverse proxy".
You can use any 3rd party DNS provider to do what you require (providing they have the necessary flexibility in their control panel).
Although if you use a 3rd party DNS provider, as opposed to your webhost's DNS services, then you must do everything yourself. Your host will configure many records for you to ensure your website/email works without you having to do anything. For example, if the IP address of your server changes then your webhost will usually handle these changes for you - ensuring no downtime.
To use an alternative DNS provider, you need to change the
NS (NAMESERVER) records at your domain registrar, to point to the new DNS provider. And then manually create the necessary
CNAME records at the new DNS provider to point back to your webhosts server. Together with
MX records for email etc.
Note that when you change
NS records (in fact, changing anything with regards to DNS), it can take time for these changes to propagate around the internet. For this reason it is often advisable to reduce the TTL (time-to-live) value on the DNS record some time before you change it and change it back once the value has propagated.