I think the reason for the inconsistencies is because most of the clone wars series is based around Anakin and Obi-wan. Rex is loyal to Anakin but as we saw when the other jedi took over in season 4, he will serve the republic first. Cody is friends with Obi-wan but he keeps to the code and still stays strict. Even in this show you still see clones complain about how the jedi operate. Only clones that i see make major emotional changes is the 501st (anakin's clones) and thats probably cause Anakin doesnt follow the jedi teachings and creates attachments.
As for Anakin being a better leader in this series, I think it fits with the movies perfectly. In episode 3 Anakin is put on the jedi council, and dont think the rest of the jedi would agree to that unless anakin at least showed some leadership skills. Also war is Anakin's element so theres not reason for him to whine compared to him in meeting.
-side note I believe Captain Rex is next to Anakin/Vader during Knightfall
Your second paragraph is incorrect. Anakin was placed on the council because Palpatine ORDERED it. Not because the council wanted him there...yet. They felt he still had much to learn. Hell they didn't even make him a jedi master when he was put on the council.
Your logic is flawed.