While I don't have a 2021, I do have a 2018 which is the same generation and I can confirm that what you're trying to do doesn't work. When CarPlay is connected, you lose the ability to play anything over Bluetooth. I believe this was programmed so that the system doesn't try to connect to the phone over Bluetooth and CarPlay simultaneously, and I've seen other vehicles that do similar lockouts, but it is an oversight in a vehicle where the mechanism they used to connect the RES to the speaker system is a manual Bluetooth connection.
To be honest with you however, I'm really not a fan of the RES' bluetooth connection in general. I've never been able to use it without there being interference in the audio connection. Maybe mine has just never functioned well, but it has never been a good experience. Instead, I have my kids use wireless and wired headphones from the various seats when I use the RES.
Ford made a good call with the 2022 Navigator, (and I would expect future Expeditions). They've replaced these terrible RES systems with 2 much larger displays running Amazon's new Fire TV Auto system. Which means each screen can function as its own Fire TV independently of each other or in sync if desired, and they have access to all major streaming apps directly. It's the 3rd instance I've heard of with the system, behind Jeep's new Wagoneer/Grand Wagoneer and the 2022 Chrysler Pacifica. That will be a major improvement over what we currently have.