Wish I had a simple answer. The training mirrors the threats that are out there in the streets. I've traveled to many countries and I agree that police in the US are more wired and on edge than places in Europe and Asia. Most of those countries don't have as strong of a criminal element with the wide access and availability of guns. I just don't get the feeling that cops are targets as much as they are here in the US. On the other hand in places like Brazil and Mexico cops seem more on edge and almost all carry their guns off-duty, and for good reason. The whole gun debate in America is another topic, but IMO goes hand in hand with the current climate of crime and policing. For a 1st world country we really do have a culture of violence here that I can't really explain.
Re the bolded part, It's usually during planned major protests/demonstrations where police are geared up like the military. When we are notified of a demonstration in our area, we're told to "gear up", which at my Dept really only means getting a helmet and baton. Every dept has the handful of officers who look like Mr Gadget with their 2 backup guns, tactical vests, tactical holsters, extended magazines, multiple flashlights, and we all make fun of them for looking so ridiculous. But again, even equipment and gear police departments utilize is in response to certain events. You can read or watch a youtube video of the 1997 North Hollywood Shootout where 2 men armed with "more gear than the military" including bulletproof vests etc went on a shooting spree against LAPD, who were only armed with their handguns. After that event every patrol car required a shotgun or other long-gun and every officer is trained in using it. Gone are the days a unit would be sitting ducks waiting for SWAT to arrive.
Also mass shootings/attacks like Columbine, Orlando nightclub, Colorado movie theater, etc and the current ISIS-inspired attacks have changed the game. Before Columbine, police were trained to set up a perimeter and hold it until SWAT arrived. They came under heavy criticism for it and rightly so. If there's an active shooter capping off people then we are now trained to go to the threat and stop him/her. I can tell you that I have had heated discussions with my own colleagues on what we would do if there was an active shooter. Not everyone agrees that we should go in and stop the threat, as discomforting as it may sound.
