I'll define a clinic as where you go for regular checkups and minor ailments.
For checkups: A few quick measurements can tell if something is wrong now, and putting them in context of your history can tell is something is starting to go wrong. And that's before the quick talk to see if there's anything obvious going on. If you can't remember anything apropos, you're good to go for another couple of years (barring slips on the ice).
If something is amiss, chances are you notice and complain about something related to it. You may not know why your hands are swelling, but you're pretty sure they didn't do that last year. You have a construct for how your body is supposed to work and you can recognize deviations.
Mental health seems intrinsically more complicated. I gather there are some simple things that let experienced psychiatrists spot some kinds of major problems, but I suspect that if they were as widely recognized as high blood pressure is, a lot of patients would hide them. They'd be "symptoms" relative to a culture, too, and not universal like mm of Hg.
But let's stipulate that some well-advanced problems are readily detectable, and others can be determined with some effort using interviews with the patient and with friends/family. What about the rest?
Over the years a number of acquaintances have surprised me when they suddenly left a spouse or quit a job. I do not say friends but acquaintances, because the mental clinician will be in pretty much the same situation. He is not a close friend of the patient, does not get to see him react to everyday stress, and only knows what the patient choses to tell him.
What will the patient tell him?
Maybe the patient will be worried: "My hands are raw because I keep washing them." Or "I can't seem to make friends."
If the mental problem is one of perception, he may think the problem is with someone else. Or that there is no problem; what he is doing is quite reasonable under the circumstances. Only talking with family and friends will tell the clinician that something is wrong; our hero won't know. And if he knows and is ashamed he may not want to tell anybody. When your perspective is distorted, your model of what is normal operations doesn't do you as much good.
And, of course, our hero may not have any local family and no friends either. Which might be a warning sign. Or not.(*)
Some people will come to the clinic but many, likely some of the ones who need it most, will not.
You can't compel ordinary people to go for a mental checkup. Or perhaps more accurately, I don't see any good ways to do that which aren't open to abuse. If Joe has a history of problems, yes. Otherwise, MYOB.
If the services are subsidized, the clinic will have a waiting list of lonely people who just want to talk. I've never manned a suicide hotline or a late-night DJ slot, but the received wisdom is that both get a lot of phone calls from lonely people.
I have a little suspicion, from dealing with some counselors over the years, that much of the "treatment" for many of the "clinic-level" problems involves simple advice consistently applied. The consistency is the hard part; somebody has to be there to observe and remind the patient (who will need to be very patient with the reminders!). When we're all urged to leave home and have our own apartments, that's problematic and pricey.
All in all, it doesn't look like there's a close analog of the physical medicine clinic in the mental health field.
However, that doesn't mean there might not be resources. Maybe not so many for Americans, living alone and not talking to the deacons much. But if you talk to your grandmother regularly, and to your cousin's friends when they come over, and look for some spiritual direction--I'd think some of the smaller problems could be dealt with and large ones spotted. Not that they'd know what to do with big problems; just know that something's wrong.
UPDATE It was pointed out in a private communication that there are clinics that serve existing patients, presumably relatively inexpensively.
Not everybody who sits like a lump at the party has some mental disorder.
And then suddenly the goblets leapt and rattled on the board and the great table shook, for the friar had brought down his huge fist like a club of stone, with a crash that startled everyone like an explosion; and had cried out in a strong voice, but like a man in the grip of a dream, "And that will settle the Manichees!"