They presented nothing of the sort in this video, and even hesitated to tell the reason of the arrest.
I come from a post-communist country, and this video sends chills down my spine.
No, even if they don't have any of the bases for a no-warrant arrest, they don't have to present an arrest warrant. They don't even have the warrant with them, the arresting officer only has to have knowledge of the warrant’s existence.
EDIT: they don't need a warrant if they have reasonable suspicion or doing a "wellness check"
Why ? Because, so the claim goes, following the law they wouldn't be able to help.
The problem they're facing is that nobody wants their help. They claim they're psychologically ill and that allows them, with help from police, to enter your home and seize you. Of course, you'll be expected to pay for this "service" (which is "not" why they're doing it in the first place)
Wellness checks are usually done by polcie officers alone, not social workers, and it's not like the rule was lobbied for by social workers, so there is no even remote sense in which it is accurately described by your TLDR that social workers got themselves and exception to the Constitution. If anyone got themselves and exception, it was the law enforcement establishment, police and prosecutors.
And, of course, they charge for this. Thousands of dollars. I don't know if the police charge for this as well.
As to who lobbied congress for this right, I can't claim any expertise. I hear it's state law, so it wasn't even congress itself.