I think this idea perfectly sums up why this is so concerning.
I don't think I have ever sent out a welfare check for a no-show. In some clinics a significant portion of my panel experiences chronic suicidal thinking. If there was a very compelling reason, such as a family member calling concerned earlier in the day and a no-show later that day, then sure I would call a welfare check. But if the person is chronically high risk (the kind of person who very well might screen positive on these brief screeners) I would need more than a no-show where I could not immediately reach them to call a welfare check. It's worth keeping in mind that a welfare check is not really benign. You are sending police out to check on someone they will understand to be mentally unstable and suicidal. This can lead to upsetting interactions for the patient or to unnecessary forced hospitalization (not to mention wasting the police department's time). In very rare cases it can lead to worse than that, like a violent interaction that really wasn't needed.
You mentioned "concerning SI," and by that I presume you mean that in cases like chronic SI you would document why you do not view the person as an imminent risk. But when such a patient does eventually complete suicide or make a seriously injurious attempt, there are plenty of experts who would make the case that the suicidal expression they had just sent in to you was obviously concerning and, as you just said, the "clear standard of care" is a welfare check. And do we really want to add the documentation burden of regularly explaining away positive screeners where we could not then conduct an actual risk assessment?
The bottom line is that I think it does open up potential for liability that otherwise may not have been present, and opens up the potential for people sending routine police welfare checks out as a form of CYA for no-shows. It reminds me of the saying about ordering labs without a clear indication: it's kind of like picking your nose in public, if you find something you don't know what to do with it!