The “private” option is using either a usb-c to 3.5mm adapter and standard earbuds or usb-c earbuds, bluetooth is A Mess. Personally I still have the pair of usb-c pixel earbuds that came with the earlier pixels that I can use if I need earbuds
I’m in a similar boat to this. I still have an old pair of apple earuds I use with a usb-c to 3.5 adapter; but I also have a pair of cheap wireless earbuds I use for more hands free tasks.
For what it’s worth, I believe most of the time voice assistance support means that it has a specific button that can trigger the paired phone’s voice assistant. I’ve never seen a pair of Bluetooth earbuds or headphones that could actually connect to the Internet and use the voice assistant without being paired to a device, meaning you should be fine if you already don’t use the voice assistant on your phone anyways.
I like Sony sound so I used MDR-XB55AP EXTRA BASS for wired for quite some time since I didn’t want the invasiveness of bluetooth buds.
Eventually, I wanted bluetooth earbuds for some situations and bought Sony LinkBuds S (dropped one, broke it) and then WF-10000XM5 for bluetooth (haven’t broken when dropped).
My requirements for bluetooth earbuds were that I could 1) confirm they were rotating their MAC address, with each connection, 2) I didn’t have to log into an account to activate them, and 3) I could deactivate a voice assistant. Sony is compatible with these requirements. I put the earbuds in a faraday laptop bag with my 4a running a MAC address reader app (to block any other bluetooth signals) across several connections and confirmed that, yep, Sony earbuds do rotate bluetooth MAC addresses each time. No online account is needed to activate or use. I have the Sony app alone in a user account on GOS for firmware updates (needs Play Services) and I don’t connect to it from home. There is an easy setting for turning off voice assistants.
Downside is I cannot tune the sounds with the new Sony equalizer algorithm option (name?) but I like default Sony sound out of the box.