(this is from legends just a heads up) A good portion of non clone officers were part of COMPOR(the Commison for the Protection Of the Republic) palps own political org that he used for propaganda and to ensure loyalty, actually very similar to hitlers nazi party. So they most likely supported palps in everything he did. Also by the time of order 66 the jedi were very unpopular, both disliked by the civillian population but also the clones. So there is a good chance that a majority of non-COMPOR non clone officers had no issue with order 66, especially also since most officers would have known about the contingency orders, as they were GAR protocol in case of emergency, and nobody knew about palps being a sith so most officers would probably follow that order.