If it's got the RCD510 with Dynaudio, the Bluetooth isn't actually built into the head unit itself on most cars it relies on a separate Bluetooth module (usually located under the passenger seat, next to/near the amp depending on spec).
The fact the phone button just mutes the radio is a good clue that the car doesn't currently have a working phone module fitted or it's not communicating with the CAN system. The Bluetooth audio menu can still appear on some RCD510 units even when there isn't a module connected, which is a bit confusing.
The module should normally show up on a scan with VCDS/OBDeleven as well, so that would be worth checking. If there is no module under the seat other than the Dynaudio amp, chances are it was never fitted or has been removed at some point.
The good news is it's a pretty common retrofit. A genuine VW 9W7 module (or similar) will give you proper hands-free and Bluetooth streaming, and it integrates nicely with the RCD510/MFD. Just make sure it's coded correctly afterwards.
Your phone being "too new" is very unlikely if the module was present and working you'd normally at least see a VW/PHONE Bluetooth device to pair with. Sounds more like the car is missing the hardware rather than a compatibility issue.
Be worth you reading this topic if you fancy changing your headunit.
https://www.golfgtiforum.co.uk/index.php?topic=290235.0