Will the military distance themselves from the President if they prove they are not capable of fulfilling the role of Commander in Chief?

The military does not get the option to determine who is and is not the Commander in Chief. That is determined every four years.

I've spent more than 18 years on active duty so far, and I've never received an order directly from the president. Neither have 99.9% of the other people in the military. Someone who outranks me (by usually a little bit) tells me to do something, and I do it. Same with those to whom I give orders. Whoever holds the office of the president doesn't really come into our day-to-day considerations, except for people at the very top of our ranks.

The President has become largely a figurehead in recent times. Inasmuch as the functions of all three branches of government (Executive, Legislative, and Judicial) have been absorbed by the bureaucracy (which includes the bureaucracy at the Pentagon), they can absorb any attempt at interference by the President, Congress, or the Court like the tar baby in the Uncle Remus story. Attempts to fight the bureaucrats will only mire the President in an unresponsive quagmire.