You never land with a tailwind and if the wind shifts the airport simply change the runway to the opposite one. If you use a circular runway and landing with a cross wind you would during your landing roll be subjected to a tailwind.
Sure, you could stop using that part of the runway but then we're back at the practical economical argument for this design.
"Sure, you could stop using that part of the runway"
Would you have to stop using that part of the runway, or stop using it within a certain height?
That is to say, while landing, if at that part of the runway the aircraft was already solidly on the ground would it still be a problem? Likewise taking off; if the take off point was simply well before that part of the runway, would that be fine (have to think it would be, because after taking off the plane has no reason to follow the runway - it's already gone)?
If that's fine, then would you actually have to stop using that part of the runway? Could you just rotate the landing/launch points so that nobody was landing/launching at that part?
Would there be taxiways to the next exit? I would have thought that a level interior ring of the runway would basically be one big circular taxiway; entrances and exits would basically be lights and markings, and planes would get directed to the appropriate entrance/exit depending on traffic, weather and the plane's own characteristics.
Sure, you could stop using that part of the runway but then we're back at the practical economical argument for this design.