Middle East - Amman-Tel Aviv

I am contemplating a trip with the family (wife, 9yr old, 2yr old) to Israel. The flights to AMM seem a bit cheaper and it would give me another country to visit.

Questions are:

How easy is it crossing the border between Jordan and Israel with an US or EU passport?
Can you rent a car and drive from Amman to Tel Aviv or is that not advisable? I have driven all over the world, so I am not a timid driver. Just wondering if agencies let you cross into Israel and how Israelis react to cars/people from Jordan.

Thanks in advance

You cannot drive a car rented in Jordan into Israel and vice versa.

There is bus service from Amman to Israel

There is bus service from Amman to Israel

It's not this simple. First, you'll have to take a bus from some central depot in Amman to the Allenby Bridge, then you'll have to go thru the lengthy formalities on both sides and security on the Israeli side, and then you'll have to take another shared taxi (sherut) into Israel.

I would concur that flying into Amman and transiting to Israel would not be a fun holiday experience.

It can be time consuming, dusty, two countries security services are involved, English is "sort of" spoken, and pending on the time of year the crossing can be as much has 45 degree (well over 100).

As was mentioned, no cross border car rentals, in fact, you will have to transfer everything from vehicle to vehicle.

Magnificent area (excellent geological hoodoos) but not a fun excursion.