This isn't the best spot for Private Relay discussions—this forum is for app developers to discuss technical issues within their apps with the Apple app development support folks, or how to implement apps for specific purposes. If you're not working on an app and its source code or such, the Apple Support Community forums are a better place for the questions.
Here's an intro to Private Relay: https://developer.apple.com/support/prepare-your-network-for-icloud-private-relay
iCloud+ Private Relay suppresses website access to the originating IP address. Some websites try to use that originating IP for user tracking, or for other purposes including fraud prevention and DDoS protection and for managing geo-restricted content, among others. Those websites can potentially get tangled and block access.
Private Relay targets the shuffling of IP addresses of HTTPS-related protocols, and does not effect DNS translations.
I would not expect a "DNS blockage" here, either. (Whatever that is.) DNS either gets translations, or it doesn't. If it doesn't, the user usually gets 30-second pauses inserted, and no connectivity. But then I've also met some very dumb and very buggy routers, too. While you're accessing your router, access its firmware and update that firmware from the vendor if newer is available, as generic advice.