Plenty of Bluetooth adapters are designed for a specific headphone model and sit flush against the cup without additional wires. When they're attached, it's easy to forget that they're not built in. See https://thebtunes.com/ for example.
A huge caveat is that if the headphones have ANC, you now have two sets of batteries and two power switches to worry about, so they're a lot less seamless.
If you have a good pair of wired headphones it's actually quite a nice compromise, because it lets you use them at home without being stuck in your chair. I'm using a Bluetooth headphone amp right now, it's very light and fits in my pocket, which is more than I can say about my laptop.
An adapter like that is yet another thing I need to put in my bag, and lacks at least some of the convenience that comes with having headphones that just wirelessly connect to my phone natively.