I'm suggesting how they would have to function in order to be able to function. The very fact that they can act above the law as they see fit means that they'll have had to demonstrate the ability to not misuse that power. "As they see fit" doesn't equate to "whatever they feel like." Any government that gave that sort of power to anyone who didn't use it as appropriate would be completely and utterly insane, and places like Noveria certainly wouldn't accept Spectre authority. You mention the word "duty" - they have a duty and if they didn't perform that by doing what is necessary - and didn't have the ability to judge what's necessary, then they wouldn't be fit to do the job.
The alternative is for them to have no responsibility, and that simply wouldn't work. The Council would not give Spectre authority to someone they weren't certain wouldn't abuse it.
No, you're still just imposing your morals on them to dictate how they must function. I'm not disagreeing with you on how they should function only telling you how they actually do function within the lore. They do not have such moral restrictions placed on them. They're free to do whatever they want, they're extremely "big picture" focus. "As they see fit" does equate to "whatever they feel like" for all intents and purposes. Again this is part of what people in the universe dislike about the spectres, even the head of C-Sec has issues with the spectres on these very grounds.
Their one and only responsibility to maintain galactic stability. That's it, that is their only mandate. It is vague enough that it could easily encompass anyones moral beliefs and interpretations. Destroying the rachni maintains stability. Saving the rachni... well, if you want to say that no species should be completely eradicated then you can say that saving them maintains stability in that sense. It's not like the council will approve of either decision, you're either a monster for killing them or a monster for releasing them. It is still your call to make - if the council really felt it was out of line they could had revoked the spectre status. Hell remember they'll let you keep your spectre status even when you're working for Cerberus, a known terrorist organization! Spectres are above the law. They have to do something insanely bad to get any fingers waved at them.
Remember the Spectre from LotsB blew up a damn building full of people and works with the shadow broker because the ends justified the means. You do some favors for them SB and in return he shares information that will save a lot of lives. A few sacrificed for the greater good of the whole, yadda yadda. This is the kind of **** a spectre has the authority to do, it is within their right legally speaking. That, again, is one of the reasons they're controversial in the story.
As for the council not giving the position to people who wouldn't abuse it... eh. You wouldn't give someone complete and total power to do whatever they wanted if you didn't want them to be able to abuse it when needed. Just saying. This is not the justicar code, its spectres. They are given complete freedom to achieve their goal no matter the cost. Very 'big picture' focused, again. "Because of the rigorous selection process, Spectres might sometimes use unorthodox methods but they rarely go rogue."
Take a look at the footage of the first spectre:
"I knew they would survive. Probably." Yeah, real high-class boy scouts, those spectres. Look, I'm not saying what you impose on the spectres is wrong. I actually do agree with you that there is a way they should behave. What I'm saying is, however, that there are no moral guidelines to a spectre. There is no real 'way' they must behave or act. They can go around and shoot people at random if they really wanted to and as long as it wasn't something huge to draw a lot of political attention the council is unlikely to even question it - the less they know the happier they are, they don't even want to know what you're doing. You're free to do whatever you want. Even when the council is giving your quests in the first game they make it clear that you're a spectre and free to do whatever you please - they're just making suggestions.