Afaik the idea behind elected prosecutors and judges is to make sure that the law represents what the people want, since a prosecutor or judge that fails to convict when the people want a conviction won't get re-elected.

So yes, they are politicians, and I'm not convinced that's a good thing.

Of course, being north of the border, I'm not exactly versed in American state politics.

But isn't it up to the legislative branch to make/amend laws? Why elect the judiciary branch, which is only supposed to interpret those laws?

This is actually a big deal in America and a fundamental change in how our government operates. Your idea is right, but since the 1960s judicial authority has grown in a way that increases their responsibility to the point where they're almost legislators themselves.

Because there's a lot of different ways to interpret laws, judges have a fair amount of flexibility in sentencing, and elected judges aren't by default just cronies of whoever appointed them.

I think prosecutors fall under the executive, not the judicial, branch

I believe that's the idea, yes. However it's very easy to see that it doesn't actually work that way. Unseating an incumbent candidate in local governments is damn near impossible, barring "scandals" or obvious extremes (but not even then, depending on the area we're talking about).

The trend to elect anyone and everyone was part of the American Progressive Movement backlash against the spoils system. Under that system every job in government was given to a crony of the chief executive.

Unitary executives and plural ones each have their pros and cons, but on the balance given the limited attention span of the electorate, I think unitary is better.

Compulsory prosecution puts all the power in the hands of the lawmaking branch. In the USA we have a traditional separation and balance of powers. Prosecutor's discretion is considered one of the balancing forces.

Yes. In the US, members of the judicial branch are either elected or appointed (by politicians). The rules vary by state and by position and both have problems. Right now there are several areas where there are open judge seats but any election of new judges has been blocked (I think PA has/had this problem). Either way, prosecutors and judges are beholden to politics in order to advance their career path and/or even keep their current position

Well I live in a country with non-elected attorneys and they become marionettes of the politicians quickly. As long as they do as the regime is saying they will be there and they don't care.

I have read that judges have to campaign and take in significant campaign contributions. Take about being neutral and only bound by law.

It makes them exactly politicians.