Hi AlienMindTrick,
Welcome to Apple Support Communities.
While we are certainly not experts with Cloudflare, it may not necessary to run both at the same time. The resource attached to your post has some great info about how iCloud Private Relay protects your privacy: About iCloud Private Relay
"How Private Relay works
Normally when you browse the web, information contained in your web traffic, such as your DNS records and IP address, can be seen by your network provider and the websites you visit. This information could be used to determine your identity and build a profile of your location and browsing history over time. iCloud Private Relay is designed to protect your privacy by ensuring that when you browse the web in Safari, no single party—not even Apple—can see both who you are and what sites you're visiting.
When Private Relay is enabled, your requests are sent through two separate, secure internet relays. Your IP address is visible to your network provider and to the first relay, which is operated by Apple. Your DNS records are encrypted, so neither party can see the address of the website you’re trying to visit. The second relay, which is operated by a third-party content provider, generates a temporary IP address, decrypts the name of the website you requested and connects you to the site. All of this is done using the latest internet standards to maintain a high-performance browsing experience while protecting your privacy."
This appears to be similar as what is posted on the Cloudflare site: Is the 1.1.1.1 app a VPN? – 1.1.1.1
We hope this helps.
Cheers!