This is quite popular in europe and is a result of lesson learned from the nazis
Well, you can. You're allowed. But at that point I decide to stop engaging. So good day to you.
Quite the contrary, just look at ADL or AIPAC.
It seems you think Israel has direct finance or control interest in AIPAC, do you have something to support that?
AIPAC is nominally funded by "individual donations", but the problem with that it is fairly trivial for a nation-state to fund things in this manner, so it doesn't really tell us anything. And then, of course, there's the question of personal connections: if you have a private person donating huge amounts of money to AIPAC, and they just happen to be heavily involved with the governing political party in Israel, I would consider them an agent of the state of Israel ipso facto even if they don't have such official position (just as I would consider, say, one of Putin's pet oligarchs a Russian agent if they donate money to an American political campaign in their private capacity).
The discussion here is about the actions of Israel versus Al Jazeera, not a possibility of banning Al Jazeera in the United States or maybe Israeli viewpoints.
Also, I am pretty sure Israeli spendings to push Israeli narratives in the US are minuscule, especially compared to Qatar's.