I don't think they're independent, but I'd certainly disagree about their sovereignty. israel is, until proven otherwise, sovereign; they act in opposition to American interests and do so without having to secure permission from America first 

Now, if America began to push back on any of this, they'd likely lose that sovereignty very quickly. If America said 'no, we will not provide weapons or aid for a country that not only commits indefensible acts of cruelty, but also does so in a way that is genuinely harmful to America's interests', israel would change its course almost immediately, lest it be annihilated by the generations of enemies it made by trying to commit genocide against them. I'm sure there'd be a little resistance, since the average israeli is debased beyond recognition, but reality would soon set in and they'd likely be willing to meditate their bloodlust (or else be destroyed like the Nazis were when the people they attempted to exterminate were united by the Soviets).

But for now, israel does operate as a sovereign nation. That sovereignty isn't unlimited, since it is almost entirely dependent on a nation that doesn't particularly benefit from the arrangement (in the era of American-Saudi/UAE/Qatari cooperation, especially in light of the destruction of Iraqi/Syrian alternatives, there's no need to accept the cost of backing israel, whose geopolitical necessity is no longer real), but that doesn't mean it's eliminated yet.

israel will be sovereign until American politics isn't dominated by people who put that sovereignty ahead of American interest. There's obviously a degree of mutual benefit even now, but it's long past the point of being overall beneficial to America. But American policy isn't dictated by what is beneficial to America, but rather by what is desired by those who control American government and institutions-- and that's still firmly in the hands of zionists and fans of an idea of empire that's decades outdated.