And this is where comparing countries gets complicated. Because it sounds like in Alaska, the Attorney General is appointed, just like at the Federal level, whereas in Nevada the AG is an elected position. So obviously in my state, the AG is more politically motivated than in yours or a 'State' Attorney in Denmark. That makes any sort of comparison hard to make.
But in even when DAs and AGs are appointed, rather than elected, there is always the chance that they have their eye on an elected office down the line, a judgeship or a legislative seat, and their record as an attorney is definately going to play into that, so the playing of politics occurs even when they are theoretically outside of politics. I would presume that to be true for any suitably ambitious person in the "State" or "government" level in Denmark or the UK, too. I won't claim that it IS true, since I don't know, but if it isn't, I'd like to hear how they prevent it.
As for brawling, heck, this thread is downright civil compared to most politcal threads in these parts.
I see more problems with DAs and AGs being elected than appointed. If someone runs for a position that enables you to prosecute, we run a very real risk of pandering to the lowest instincts found. An agitated population in the heat of the moment could easily impose way too strict or unfair laws by electing the guy who promised to strike down the hardest.
There will always be ambitious people, regardless of what country they come from. I've never heard of a Danish DA or an AG seeking political power, though. Danish politicians have a wide range of educational backgrounds, but they most certainly aren't generally lawyers, like in the US! Here, civil servants tend to stay in that world. At least they don't get kicked out of office at the next election.
And I really don't think you want to compare playing of politics in Denmark with playing of politics in the US....