Exactly. I routinely switch my headset from my laptop to my phone and vice versa. Doing this with a jack is trivial. Doing this with a BlueTooth headset is very annoying and requires additional steps on both devices.
Anecdotally, my experience with Bluetooth headphones is that they’re easier. I don’t have to do anything more complicated than you do — if I turn on my Bluetooth headphones in range of my computer, they connect. If I turn them on in range of my iPhone or iPad, they connect.
If any of them start playing sound, it comes out the headphones. No cable twiddling required.