In theory a perfectly reflective silver object would never cool down and a perfectly black one would cool to -270C however much power you dumped into it
And a perfectly blank one will absorb light, heat up and also emit blackbody radiation.
A perfect emitter facing an infinite cold sink is in perfect equibirium at the temperature of the cold sink.
You said: "...would cool to -270C however much power you dumped into it" and that is not true.
And in any case you don't need to be a perfect emitter to be in equilibrium with the temperature of the cold sink, all you need is infinite time, but any emitter, no matter how bad will eventually equilibrize.