You are ignoring the other ways heat dissipates.
Okay, sure, convective heat transfer could definitely overpower solar radiative heat transfer but you'd need a fairly strong wind to get to the point where a white and black object will have essentially the same equilibrium temperature. And I doubt conduction has any significant effect in this situation.
Also, once sunlight has heated the object, the radiated heat is at much lower frequencies and the emissivity is more or less the same then (this is somewhere on Wikipedia).
Actually it's quite the opposite, as the object's temperature rises the frequency distribution of the blackbody radiation shifts toward higher frequencies. This shift toward higher frequencies is what causes the increase of heat transfer out of the body via blackbody radiation. If it were the opposite, and the frequency distribution moved to lower frequencies as the temperature increased you might have bodies near absolute zero giving off high energy radiation. It's true that emmissivity will change as a function of temperature and for most materials the emmissivity will tend to increase with increasing temperature, but you're not going to see a white and black object achieve the same emmissivity just by exposing them to 0.1 kW/m^2.
It is a fact that brighter surfaces reach lower eq. temp., not just because the day is too short for them to reach it.
I'm a bit confused here, maybe I'm misreading your statements and if so, I apologize, but you seem to be contradicting yourself.