Contrary to the trope of people being 'in therapy' for decades, I'm pretty sure that most counseling psychologists want to help people get through a difficult time and move on successfully. At least in my experience, they don't assume that the default human state is being stuck somehow. They seem to assume that normality is getting things done and making progress toward whatever one's goals are. They're trying to put another shoulder to the rear bumper to get your car out of that sandpit, so you can drive on. Then they'll wave goodbye.
The two counselors I've talked to seemed to believe so strongly that success is the default, that if someone was stuck with a long-lasting problem, they would suspect that the problem wasn't really a problem: it was probably actually a solution. Not necessarily an optimal solution, but a solution that worked for the person, a way they'd found of making their lives manageable. So some of the counselor's work would be in helping the person understand how and why they themselves were at least to some extent working to maintain their problem, because at least in some ways that problem was a working solution to some other problem. If the person wasn't really happy with this situation, then the counselor would try to help them find a better solution to the second problem, so that the first problem would become unnecessary, and thus a lot more easy to solve in its turn.
I'm sure it can be very rewarding for a counselor to help someone through a process like that, but then it must also be miserable to have to deal with someone who doesn't actually want to find a better solution, doesn't want to change their lives, but just wants to feel better about the lives they have. I think that wanting that is pretty much like just wanting a drug. I think most counselors are trained and motivated to be like surgeons, trying to fix things, not like palliative care specialists, who are only there to ease chronic pain until the end. So I think most counselors are indeed likely to be depressed about clients who don't actually want to change anything except their own feelings.