Does the Jedi Council wield higher military authority than the Supreme Chancellor?

Question

When General Grievous' location was discovered in Episode III (0:50 of this video), Anakin relays Palpatine's request for him to lead the campaign to capture General Grievous on Utapau. Mace Windu disagreed saying, "The Council will make up its own mind on who to go, not the Chancellor."

I had believed that the office of Supreme Chancellor is the commander-in-chief of the GAR in the same sense as the POTUS is one. Since the Jedi Order is meant to be subject to the authority of the Galactic Senate and its offices, why did Mace Windu say what he did?

I had considered three possible answers to my question, although I can be wrong on all counts:

1. My belief about the Supreme Chancellor's degree of military authority is wrong.
2. This is a battlefield decision. Normally, generals should only make battlefield decisions that defy higher authority in contingencies where it is critical to do contrary to what the orders say and there is not enough time to update and consult with said authority to change orders. Can General Grievous' location be considered a valid contingency to defy the Supreme Chancellor over, though?
3. The Jedi Council wilfully defied the Supreme Chancellor because they distrust him and suspect him to be under the influence of Darth Sidious (they have never suspected him to be Darth Sidious himself until Palpatine revealed it on his own), even though they have always been lacking in evidence.

Please share your insights on this issue, thanks!

Answer 1

My existing tentative answer got deleted for whatever reason, but based on its feedback and some further reading here on SE, I've formulated a better one anyway.

First of all, the top-voted answer on this question indicates that the Jedi are essentially consultants where official Republic structures are concerned. They may fulfill roles in the Army, but ultimately they are under the authority of the Council, not the Republic per se. The other answer also suggests that the Council are an authority unto themselves, so I think we can assume this.

This question and its answers also indicate that Jedi were brought in as Generals outside the normal procedures, and were very valuable to the Republic, which would explain why the irregularities inherent in this situation would be tolerated by the Republic. Of course this was also an element in Palpatine's plan to scatter the Jedi so that they'd be easier to destroy.

Given these two points, my informed but not directly canonical guess would be that since the Jedi have their own power structure (that works with the Republic, but isn't subject to it), they can be asked to function as generals in the GAR, but are not obliged to accept such commissions. In Anakin's case, since the Council has direct authority over him but the Republic doesn't (i.e., he's effectively a civilian where the Republic military's authority over him is concerned), in effect the Council is not refusing the Chancellor's order; they're not even officially involved. Anakin was requested to lead the operation; the Council are ordering Anakin to turn down the Chancellor's request, which as a Jedi, he can do. Whether or not that refusal is his own decision or the Council's is irrelevant where the Republic government is concerned.

Answer 2

The Jedi are not a branch of the military, and are therefore not under the authority of the commander-in-chief of the armed forces (except insofar as many Jedi are also generals in those armed forces and are thus individually in the chain of command). Presumably the emergency powers decree that the Senate passed in episode II did not include the transfer of its authority over the Jedi Council to the Chancellor, only its authority over the armed forces.