My experience with Bluetooth headphones was that on an iPhone they were occasionally annoying, while on Linux they were quite poor and on Windows they were more like a practical joke. This would have been like 4-5 years ago though, maybe all the stacks have been improved.
Second thing: The actual default behaviour seems to be to use the profile that was used last time. But for me, whenever MS Teams is running, it decides that the correct thing is to switch it to Headset mode, because Microsoft knows what you want better than you do, and you want the option that produces shitty sound quality, but bidirectional audio.
It's just never going to get priority because it doesn't have the marketshare. Sony doesn't care about Linux and Canonical doesn't care about Sony.
If I were you, I'd consider an external bluetooth adapter that supports A2DP and connects to the 3.5mm plug on your computer, like this one: https://aluratek.com/universal-bluetooth-optical-audio-recei...
Look for AptX low-latency support too if you care about audio lag (for movies or games).
Everything switches seamlessly from LG earbuds to Samsung Galaxy A52 5G, Nintendo Switch, Envy with Pop OS and Windows 11.
I have no idea what people are doing.