Wow so they were kept over two weeks while not showing any symptoms?? That's very interesting...
Well, maybe not that interesting. Today it's very hard to remain hospitalized for 2 weeks when actively symptomatic, but not so back then. Also, the ability to fake symptoms isn't so unique to psychiatry. You could fake your way to a pair of glasses. You could fake all sorts of pains and neurological symptoms that would lead to extensive testing.
Which brings us to your video. There are tests (various brain scans for example) that show some biological process involved in mental illness, at a population level. These scans are not useful for any individual patient, but differences are still seen when you average many people together (suggesting lots of overlap between those categorized as having the particular psychiatric disorder and those categorized as not having it, yet a difference existing nonetheless).
Even with such tests, the question is rather loaded. It implies that without a blood test or radiological test, something doesn't exist. We can show the existence of many things without such 'biological' tests. Many psychological tests, for example, are rather rigorously studied. Many other conditions people tend not to doubt, like migraines as already mentioned, don't have diagnostic tests, so the whole basis for this question seems misguided.
I'm curious to know how much editing was done in that video. The psychiatrists may have added more than what we saw.