Well, I can't speak for anyone other than myself, but I have been a headset user since the mid 1990s, when I was in inside sales, and I won't talk on the phone without one. I agree; there is no substitute for DECT; BT is a poor second choice.
The point is, as you've confirmed in your reply, is that the OBi ATA only connects via the RJ-11 analog telephone interface. Headsets interface to the handset audio signal on the attached telephone.
I use two methods. I have a few DECT cordless phone handsets with phone jacks. I plug in a high-quality corded headset, clip the headset's strain relief to my shirt, clip the phone to my belt, and I can wander around. It only gets snagged very occasionally, so I don't mind this solution.
I also have a high-end/business-class Jabra PRO 9470 DECT headset, connected to an OBi IP phone. The phone and the headset both support Electronic Switch Hook, or EHS. This uses a special (and overpriced) set of cables to interconnect the two devices, to control the phone being on or off hook remotely. Years back, I had an electro-mechanical handset lifter, but those are not necessary if the phone supports EHS. The Jabra's base connects to the headset audio jack on the phone (RJ-9) and to the phone's aux jack for EHS.
The source with the largest selection, best customer support and most expertise is Hello Direct (they are owned by Jabra, which bought GN Netcom, the business headset company in Scandinavia). They also sell Plantronics and Sennheiser.
So: solution #1 is quite inexpensive, while solution #2 is much more expensive. #2 is completely cord-free on your head. Take your pick. AFAIK, the solution you imagined (some sort of cheap, integrated thingy that would plug into the OBi) doesn't exist.