I had similar problems with the Boss Waza Air, I can't remember exactly what now but it wasn't just automatic if the phone had connected to other devices in the meantime. My Sony headphones almost do the opposite, they automatically turn themselves on - I start a Teams call using my Laptop speakers and it somehow turns on the headphones that are on standby (and not on my head), my JBL bluetooth speakers have their own idiosyncrasies when connecting two of them, my car bluetooth does some very annoying auto play things and seems random about which phone it decids to connect to. So as long as the Bluetooth actually works I give these things a bit of a free pass as it just seems the way it is with Blutooth to have slightly odd ways of connecting to different devices.
These Positive Grid headphones are now on my shopping list, seem to be sold out everywhere though.