Its actually a professor named Martin van Creveld, who is a professor of Israeli history in Israel. So its not these Islamic radicals that coined this phrase, it might not be "official policy" of the Israelis, but its not some made up term by their enemies either.
He is quoted as saying
"We possess several hundred atomic warheads and rockets and can launch them at targets in all directions, perhaps even at Rome. Most European capitals are targets for our air force. Let me quote General Moshe Dayan: 'Israel must be like a mad dog, too dangerous to bother.' I consider it all hopeless at this point. We shall have to try to prevent things from coming to that, if at all possible. Our armed forces, however, are not the thirtieth strongest in the world, but rather the second or third. We have the capability to take the world down with us. And I can assure you that that will happen before Israel goes under".
As for your claim that "Israel as a nation has more than proved its responsible enough to maintain a nuclear arsenal."
They are one of 4 countries in the world that have not signed the Nuclear Non Proliferation Treaty.
I do not want Israel to fall or its people to suffer. However its not the US peoples responsibility to keep them safe, nor anyone elses. I truly believe that if peace is ever to be achieved in that region, it needs to not be meddled with by other countries.