You know that nothing about that changed in the Empire, when Imperial Officers took command. (Well, the officers weren't at the front line, exept in heavy armored vehicles.) And the droids used exactly the same tactics.
It's not the Jedi's fault that military tactics in the Star Wars universe seem underdeveloped.
By the way the Jedi do answer to the Republic. They have sworn loyalty to the Republic long ago. You remember Obi-Wan? "Anakin, my allegience is with the Republic, with democracy!"
And who placed the Jedi in command of the clones? Who did it? Not the Jedi themselves. Either Palpatine or the Senate.
It was all part of Palpatines plan. In order for him to be able to kill the jedi off without a huge public outcry he had to erode thier image to the point where thier deaths would be seen as the right thing to do. Hence him putting a jedi order with no military experiance heading a slave army and being forced to actually fight a war with it.
Just thought I'd point out here that The Mandalorians have fought with the jedi aganst the sith trumverite, again during the new sith wars, and later during parts of the second galactic civil war.