[Series 6 spoilers up to Let’s Kill Hitler, of course.]

Consider their motive in wanting to punish Melody. As far as they know, her only crime is murdering the Doctor. They don’t know how or why she did it. They just know that she did it, and that’s enough for them to immediately cease their assault on Hitler(!) and go after her.

That’s the thing. Melody is, in the eyes of the Justice Department, a worse criminal than Hitler. Just for killing one man, the Doctor. At its core, it’s emblematic of the universal Doctor-fanaticism that Moffat made some points about in Series 6 (and it’s also part of why I like the episode; the Doctor has to work against people who in any other scenario would be his absolutely staunch allies).

It’s pretty clear why Moffat chose Nazi Germany as the setting - he wanted to make the contrast as clear as possible (also have Rory punch the Fuhrer). The Teselecta crew, and the justice system in charge of them, holds so much love and respect for the Doctor that his death alone outweighs even the most massive atrocities.
Crazy? Perhaps, although remember that were it not for the Doctor, Earth and reality would have fallen time and time again. By killing the Doctor before his time, do you doom the universe? The Doctor probably does need to stick around, and has given the cosmos ample reason to love him and want him alive, but that love (and infamy) can be damaging. That’s why the 11th Doctor ultimately wants to slip back into the shadows in order to continue his work.

Anyhow…
In some ways, Let’s Kill Hitler is a story of a celebrity and their obsessive, over-protective, vicious fans. Strip back the time travel, technology, laser beams and legal-ese, and it’s not that massively far removed from the extreme behaviour of some modern fanbases! Amusing and fascinating stuff on Moffat’s part - I think the whole underpinning theme of the Doctor’s fame was one of S6’s strengths.