European police is very much armed. Also the UK has armed units even if your usual beat cop is limited to pepper spray and a baton or whatnot.

Elsewhere police regularly carry pistols, but are also trained in how to not use them. In my state there's even an assault rifle (actual one) in every police car. Decades pass without anyone getting shot.

I think it’s a blend, in my example the police would bring them into custody, and then trained people work with them after that working out what happened and working with the justice department.

Nope. Police is not trained to deal with e.g. a psychotic person seeing zombies, if they try to take them into custody they're only going to make things worse. It's fine if police are first to the scene, but they should be trained enough to a) recognise that the person is psychotic, not actually threatening anyone b) call for backup from the people in white coats with haloperidol shots and c) shoo away bystanders. Perimeter duty. Yes, after 2 1/2 years training you're on perimeter duty get used to it that's your job.

The US approach to a paranoid schizophrenic scared shitless seems to be to make it worse by laying siege and throwing flashbangs.

There are many things that police aren’t needed at, like domestic issues, but there are plenty we do need them at too.

That's probably the bulk of what beat cops are doing over here, short of investigating noise complaints on behest of the municipality and documenting traffic accidents, car thefts, maybe a break-in, whatever. Which is also why they always, and I mean always, come in male/female pairs.

The core problem is that there are so many things that can help prevent the problems from arising to begin with that need to be done before policing is even considered. Better healthcare, housing, education, etc. Police are, at best, a last resort solution to desperate cases, and they tend to be hammers looking for nails as a result. It might be possible to do it well, yes, but it's very hard, and you should really be looking for a less antagonistic solution first.

To take your idea of "speeding at 100+" as an example: This could be solved by replacing cars with public transport, such that people don't really have so many opportunities to go 100+ to begin with, or by using traffic calming techniques to make it feel too unsafe for anyone to want to try, or using alternative road layouts to make it significantly harder to pull off at all (e.g. roundabouts). There are many options, almost all of which are better – and less punitive – than the police.

Also, tangential, but...

crisis councilors aren’t going to be driving trying to perform a PIT maneuver.

Of course not; PIT maneuvers would kill people.