Eh, that's rather inconsistent. They're a private group, sometimes with ties to the American government, but often with the UN rather than that. Also, often when they were tied to the government directly (both the more recent post-Civil War period and some older storylines too, like back in the late 70s), the government ties are portrayed as something negative, with the government wanting to control super humans and having clashes with the Avengers due to that. There are standalone individual issues where they help mutants too (like Thor facing the Marauders during the mutant massacre to save Angel).
It's only when you jump to the X-Men side that you get this narrative of complicit agreement with it all, like the recent issue in the last run of Uncanny where Captain America was playing security for a mutant hate rally that attempted to lynch Cyclops, which in this case specifically just becomes ridiculous when in his own book he was standing up to very similar far right groups. Really, it amazes me that Marvel just let's the X-Men editorial shit on every other character like that.
Here's the thing about this. Are the avengers a private group? kinda/sorta, depending on the year. no problems there. But note that I mentioned several of them had high level ties to SHIELD, and even acted as director. Those individuals were:
Tony Stark: (Executive Director)
Steve Rogers: (Executive Director)
Sharon Carter*: (Executive Director) not an avengers member, but closely affiliated with Rogers
Jessica Drew: (Shield Agent)
Eric O'Grady: (Shield Agent)
Black Widow: (Shield Agent)
Wendell Vaughn: (Shield Agent)
Sam Wilson: (Shield Agent)
James Barnes: (replaced Nick Fury Sr. as "man on the wall")
Black Panther: not SHIELD, but Wakandan intelligence is demonstrably better and he's aware of everything SHIELD does
On top of this, Avengers Identicards grant everyone who carries them top level clearance in the US government and all that entails.
The idea that SHIELD was heavily involved in manufacturing Sentinels all this time (which is obvious from the start, even before Maria Hill admitted it during Bendis' Uncanny- Sentinel Squad O.N.E. was proposed by Val Cooper) and the Avengers simply didn't know about it and weren't in a position to do anything about the skyscraper sized murderbots murdering mutants absolutely defies belief. It flat out isn't possible. The Avengers were either ambivalent or complicit.