The Initiative that has a poor track record is specifically Maggie Walsh - who was civilian, not military. She performed experiments that the military didn't know about, that most of her own staff didn't know about, kept secrets from everyone and built secret facilities. This resulted in the military side being ambushed from within by an unknown experiment operating out of unlisted facilities. No-one will perform well under those circumstances.
There is nothing to show that what remained afterwards is power-hungry or arrogant; they were seen to operate as a military unit and after the events of season four a group of leaders were shown to specifically not be power-hungry or arrogant. They decided that the Sunnydale crew were competent (which they were at that point) and left them alone. The opposite of what would be expected if they were, in truth, power hungry and arrogant.
Regarding their legal standing to operate in the US, we don't know what it is. They aren't the regular armed forces when the Initiative was running in Sunnydale, so wouldn't be held by those rules. After that, they were a purely military organisation of monster hunters; essentially, Special Forces that kill things that go bump in the night. Not monster experimenters but hunters. But still not part of the regular military. Certainly, they sounded competent.
And if the end of the world is coming and someone can help, do you really turn down their potential help? That's just stupid. "No, I don't want to play with you because someone not you was mean."
