Spin-up produces only a short transient, you do not need a PSU rated as high as short transients, the 60W brick "might" be ok, if 60W is enough to meet the max sustained requirements & it is designed to handle these spikes & transients. If the manufacturer recommends it , it "should" be fine.
Way over kill! A Corsair 550W or 650W is the ideal or 750W for future CF. The Antec New True PSU's of the same wattage earn the same recommendation.
If you just want an ultimate, never worry forever, expand till Star Trek, recommendation, it would be for a Corsair 850HX or Antec Signature 850W...
With that 4870 X2 you are fully using your 620HX, but it can handle it, I would have gone a 750HX myself for cooler, quieter operation with more headroom for transients.
A 950TX would be a nice improvement & allow for a second 4870 X2.;)