You are not correct. Colonels make Brigadier General by competence with allowance for diversity and multicultural consideration. But, to move up from one-star to two-star, and especially to three-star you must be willing to play ball with the politician worms. Every three and four stars have signed the corrupt bargain.
Tell that to my now retired Brother-in-law who was a Major in the Air Force.
Not only was he not diverse enough, he also didn’t play ball with his elitist officers.