It's misleading to say that Nato or the UN should provide a secure border because what it really means is that the US should provide a secure border. Neither Nato nor the UN does anything without the US actually doing it.
There is a difference between NATO and the UN. Membership in NATO is not open to Israel's enemies for one. Asking for NATO intervention means that if Syria or Iran want to enter the conflict they would have to go through NATO first. I think that this is a shrewd calculation by Israel and the United States.