There are many systems and solutions for this kind of problem, though I don't know of any exactly as you describe (and they would be unnecessarily complex). The main options I've seen used well include devices such as Microsoft's RoundTable which selectively uses directional mics to key in on the speaker without causing feedback. This however only works for groups of up to about 12 or 15. Another nice feature of the RoundTable is that it can support video that will automatically follow the speaker and also provide a full room panoramic shot.
Beyond that, you move in to the realm of boundary mics. Boundary mics are microphones that can be placed on each table or even in front of each speaker. A well designed conference system using boundary mics should key in only on the mics of speakers and should avoid feedback problems. The mics can either be permanently installed or rechargeable wireless units that sit in a charger at the door. Good systems also have an individual mute switch on each mic.
Theoretically, a similar setup could be done with headset mics, but then you have to worry about a variety of issues including sanitary concerns, proper fitting, noise from motion, more difficulty in muting/checking mute status, etc. You can't really have everyone in the organization have their own dedicated headset since it would be difficult for the system to tell if someone is in the room or simply next door. Most such systems use a similar (to the boundary mics) charging station that activates the headsets when they are removed.