Well I dont see anything wrong in what Shalya said, he was a charioteer or should i say a forced charioteer, but it was not his job to take out the wheel.
When Karna lost to bheema, he fled away from the battle field in another chariot. He did the same in Gandharvas war. Then why did he bother to lift it now? He was the senapati, anyone could have given him the chariot.
It is because he didnt wanted to waste time and when karna came down to lift the chariot Arjuna was unconscious for few minutes but he recovered quickly than expected. Yes, Krishna did not help him in gaining consciousness. He recovered on his own. Krishna only helped him when Ashwasena entered in Karna's nagastra and Karna did not knew about it. He did not want bad name to karna, because thoughh he didnt have any idea about it people may say that Karna will never be capable of Arjuna as he did not die in his hands and his abilities may not be recognised. Also he did not want the best warrior Arjun to die of poison with treachery. So to do good to both the warriors he took this decision.
We have an entire Parva dedicated to Shalya. He too is a very important person after bhishma, drona and karna.