Pip, 
OK  I'll retract the comment about an ulterior motive, although to join OSM and make just ONE edit ever, does seem strange.
But moving onto the road I simply don't get your explanation ... granted I rode East to West, so would not have seen any pertinent signs at the West end, and of course at the East end, yes there is a No entry on the gate to another private road that leads to the estate.  
Could it be you got the roads mixed up?
I have looked of street view and while I see signs marked Hafod Estate, there is nothing to indicate its a private road. There are even "official" brown tourist info signs pointing to the Picnic area.
Furthermore, an old streetview image at the junction with the B4343 does show this road signposted as the B4574 to Cwmystwyth/Rhayader.   I cant believe its since changed.
I too am a motorcyclist and I hate to see excellent road which appears to have been recently resurfaced, not correctly tagged, and perhaps not allowing others planning routes in the area to use it.
We can continue the debate, but I suggest you have may have confused this road with the Private one (showing as a track) that parallels it to the South, just under the path labeled "Coed Hafod Walk". 
Rgds. Russ