The privacy issues come from the app, not the headphones. Don't use the app and you're fine.
General privacy rules apply: any data you provide will be stored and sold. If you tell an app your height, weight, etc, the company will sell that. That shouldn't be a surprise to anyone.
The only interesting bit here is Bluetooth tracking, meaning people can see a unique identifier by being near a Bluetooth device. This is true for all Bluetooth devices (and honestly any wireless protocol besides point to point wireless), so not unique to headphones. If your phone connects to cell towers, your headphones aren't going to be the ones breaking your cover...
I've purchased a few devices in my day that will not work without an app and/or an account. I cry for 5 minutes about what a fucking dystopian future we live in and how I've lost almost all excitement for new tech and then send it back to where it came from.
A couple of years ago, I found my old minidisc player and I've been using that for music. There's a FLOSS burning application made by the minidisc community that lets you burn discs from your browser. I've been enjoying my low tech music listening, knowing it can't be tracked by some algorithm. 💽