it's been my experience that mental health care and job search support are two distinct entities and they really don't overlap.
A number of years ago I did a job search program for people with disabilities and the program focused exclusively on job search skills - no support for anyone's disability was provided beyond guidance on our legal rights on how to request necessary accommodations in the workplace. But I was there for help finding a job - not mental health support. I get my mental health support elsewhere
Yes it can be frustrating when counsellors don't get mental health, but in the context of a job search I don't necessarily expect them to get it. I have my psychiatrist, addictions Dr., and support groups to support my mental health recovery.
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"I danced in the morning when the world was begun. I danced in the moon and the stars and the sun". From my favourite hymn.
"If you see the wonder in a fairy tale, you can take the future even if you fail." Abba