Bluetooth issue: iPhone paired but no audio output

The short version is that I can’t get audio via Bluetooth on my iPhone, although I think it's more complicated than that.

My iPhone X (ios 13.1.3) has no issues seeing or connecting to Bluetooth devices, but even though devices are showing as connected (in the iPhone settings -> Bluetooth), I still get sound from the iPhone speaker (and not the intended output). When I open the control centre (& click on the wireless speaker icon), I cannot see or select any Bluetooth sound outputs. 

If I restart my phone, Bluetooth will start working again fine - for a while. I have tried resetting the network settings but this didn’t help. 

It’s definitely the phone (not the speaker) - I’ve tried multiple wireless speakers and my car’s Bluetooth system. Curiously, I can still make hands-free calls in my car (with Bluetooth) but I can’t play music. I guess it’s a slightly different protocol? 

This has happened in the last few weeks - perhaps with an iOS update although I can’t be sure. It may also correspond with my beats headphones behaving strangely (failing to turn on, then failing to turn off, although these now seem to be working fine). I suppose it’s possible that these headphones are failing to ‘disengage’ the Bluetooth audio or something. Is that a thing?  Curiously, the Beats headphones connect for audio every time!

This is a very frustrating issue. Any fixes / suggestions? Or failing an actual fix, is there a way to reset the bluetooth module or something similar in the phone (without restarting)?