The music / voice / connect construct of the spec seems to have improved this in the past few years so that it's possible in software to connect.
I downgraded from a flagship to a dirtpile phone, and now when I turn off Bluetooth from the Android Quick Settings, it turns itself back on exactly once, every. single. time.
Sending interactions is pretty reliable too for switching between devices. Pressing the call button on my car or my headphones signals my phone to instantly change audio contexts, but it's inconsistent, and some devices send "pause" signals at will.
It's probably an accident, but my car displays correct, up-to-date, scrubbing and metadata, and I can get audio from my ANC headphones at the same time.
Bottom line, I've seen the "Connect" button work, and there was even a Quick Settings panel listing paired devices for you to plug/unplug at your pleasure, but only if you've paid for an expensive phone.