The 501 is a single pilot so will be below 12,500lbs. I doubt if this has any relevance but the C550 which is variant and believe has the same engines as the 501 can carry 14,500lbs in certain forms. So I don't think the weight may have been an issue even if it was over weight.
Of course if it was over weight, out of COG and say only idle power was available then that could be a problem..But there are still the reports of engine vibrations. Time will tell..