You're right, because the nRF52 series are Bluetooth Low Energy (BLE) devices, they can't stream music at all :-) Music streaming is done via Bluetooth Classic (as of now). BLE != Bluetooth Classic
I was under the assumption that BLE chips have regular BL support as well as a fallback for when you actually want to use the thing for something productive. Is that not the case or are most phone manufacturers just lying while putting two chips on devices?
Like if a chip supports Bluetooth 5.1 that should include both, no?
Have you tried a Bluetooth 5 LE headphone? It's a completely different radio and protocol from previous generation Bluetooth headphones. I don't know if there are any in the market yet. There certainly isn't any based on nRF5340 yet.
I don't know if the range will be better, but latency and pairing process should be much better.