JKR doesn't consider him to be incompetent:
"Dawlish had to be good. He had to be good because he became an Auror. There's no denying that. But he has his weaknesses and Dumbledore knew how to exploit them. Let's face it. Anyone, anyone going up to Dumbledore pre-trying on the Horcrux, pre-maiming his hand, anyone is gonna be in trouble going up against Dumbledore. Even Voldemort didn't want to do it. So there's no dishonor to Dawlish."
Given the quotes in the question re: confunding and that we see Dawlish engaged in capturing Muggle-borns, it isn't unreasonable to presume that he is under the charm during the entire scope of book 7.
Additionally, from the same source, JKR indicates that he is mentally and physically affected by the things that happen to him:
By the time Augusta Longbottom got to him, he had been-- several people had attacked Dawlish. I mean, I think he was a bit punch-drunk by that point, you know. He had become a favorite punch-bag of the Order of the Phoenix by then. So I don't think he was firing on all cylinders. But I really saw Mrs. Longbottom as a powerful witch.
So - he isn't incompetent. He is likely continually under mind control and thus there is no reason for the Death Eater Ministry regime to remove him when he is serving their purposes, and he is over-exerted to the point of being 'punch-drunk' - which in one definition is stated as "stupefied by or as if by a series of heavy blows to the head."