We do indeed. They were created because Republicans (and in particular, Nixon) needed a new thing to harp on for the election, and they went for "tough on crime".
Not even close. You could just as well have answered "khrushchev" with no loss of accuracy.
Attorney General’s Commission on Special Weapons and Tactics (S.W.A.T.)
Beginning in the 1960’s, local police were confronted by increasingly well-armed individuals and groups who were willing to engage in armed confrontations with the police. The traditional method of response by uniformed patrol officers placed both officers and innocent bystanders at increased risk.
In many nations of the world, such situations would likely be handled by national police forces. However, the American people have historically been very wary of deployment of federal forces within local boundaries.
It became clear that a new method of response to such complex, high-risk and often high-energy situations was needed. Such a response required expertise and weaponry beyond the normal capability of local law enforcement agencies. Thus, the concept of SWAT (Special Weapons and Tactics) was developed by the Los Angeles Police Department.
Originally, the SWAT concept was for counter-sniper and other high-risk situations that in the past would have provoked an inordinate number of shots being fired, often with injuries to innocent persons. Over the years, SWAT has evolved into the management of barricaded suspect situations, the service of high-risk warrants, dignitary protection, and the actual rescue of hostages.
Under the SWAT model, verbal techniques and physical tactics would combine for seamless management of volatile situations confronting local police. The primary purpose behind this concept was to reduce risk to the police forces involved, to the suspects, and to the community at large.
Most of these situations are resolved with verbal tactics utilized by trained hostage negotiators who are frequently an integral component of SWAT teams. Seldom are physical tactics necessary, and even then the actual firing of shots rarely occurs.
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And what are those "originally intended functions"?
I think that is answered above.
... there's no justification for having every police department, even those on university campuses ...
Whoever heard of shootings at schools or college campuses?
I'll leave it up to you to track down statistics on the practices of European gendarmes.