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Where are you using the headphones? What is your threat model? Where do you live? What kinds of people are you likely to be near when using the headphones (and, therefore, have Bluetooth on)? What kind of device(s) will be attached to the headphones?

I run Bluetooth in my car and at home. Every once in a while I have my Bluetooth headphones on to listen to podcasts/music and/or make calls when I'm out and about … but I wfh, so that's not very often 🙂

When I'm at home, there's nobody who's going to be picking-up any stray bt signals (that matter, at least), as my nearest neighbor is ~150yds away.

When I'm driving, I'm never close enough to other vehicles long enough to worry they might be searching for my phone.

When I'm using my headphones out-and-about, the odds of someone in the grocery store or at the gas station trying to find and exploit my phone are incredibly small.

I don't run Bluetooth when I don't need to, but for battery-conservation purposes, not privacy concerns (I do the same with WiFi – why run it if you're not using it).

If you're more worried, or live in a dense-ish apartment/condo/townhouse arrangement, are on heavily-utilized public transit, etc, you may have a bigger concern – but even then, the attack options tend to be pretty small, few, and far between.

The safest option, of course, is to not use Bluetooth (or WiFi) at all – but that's not really practical with most modern phones (since they don't have 3.5mm jacks any more (praise God!)).