You are making this much more complicated than it needs to be without any practical relation to what actually happens when you travel. Handing two passports to every official you see in the airport is only going to confuse people who aren't familiar with dual citizenship and waste much time.

Wrong. You don't need to present anything to leave the USA-- even if you aren't a US citizen.

Hand over to whom? The immigration police that they have stationed at the jetways of all flights out of the USA?-- oh wait, there's no such thing. The airline isn't going to care how someone "got in" to the US.

Actually, only the Israeli passport would need to be shown. The airline only cares that you will be accepted where they are flying you-- not how you got into the country of the origin of the flight.

Again, there is check-in and immigration. It is ever so simple. At check-in for your flight back to the USA, you present your USA passport to prove that you will be admitted into the USA. The airline isn't going to page through your passport to see how you got into Russia-- nor would they care.

At immigration on the way to the gate, you give them your Israeli Passport so they stamp you out of the country. You don't need to give them your US passport because they don't care if you will be admitted where you are flying to. These are two different processes with two different purposes and the two need not be confused.

Well, if you only show the Russians your Israeli Passport on the way out, that wouldn't really be a problem now, would it?