Why does Israel make its citizens do a 1000 mile bus trek? Why can't they fly and connect in Cairo or Amman or Addis Ababa where connectivity already exist? With what Israel is able to pull for its citizens for other causes, why is this seemingly problematic?
From what I was reading, Israeli Muslims have to apply for a specialized Hajj passport from either Jordan or the Palestinian Authority, as KSA will not allow anyone with an Israeli passport to enter (or actually anyone from any country with an Israeli stamp on theirs). The Hajj passport is only a temporary one, which must be turned in immediately after completion of the Hajj. My guess is that this paperwork would make a normal connecting itinerary too difficult, or the passport itself wouldn't even allow travel to outside nations.
This. It boils down to citizenship. Arab Israelis are still Israelis and have to get a temporary passport from Jordan to get into Saudi Arabia (or any country that prohibits Israelis). However, that temporary passport for hajj can't be used for air travel currently and instead is only usable for hajj on very specific bus routes. There are other categories for the temporary passport that can be used for business and family travel that I believe can be used for actual air travel but the hajj one is an exception if I'm not mistaken.
The issue isn't Israel making Arab citizens do anything. But as long as Israeli citizens are banned from Saudi Arabia, Arab Israelis are forced into a fairly complex process to perform hajj.
There was a time when Palestinians in the West Bank and East Jerusalem and some Arab Israeli citizens could actually apply for Jordanian citizenship but that was ended when Jordan and Israel normalized relations. There are still some people who managed to get a true Jordanian passport and for them travel is far less of an issue.
Even for Palestinians, performing hajj is really hard. Palestinians technically do have access to the Palestinian Authority passport but getting visas on that is extremely hard (even to Arab countries). Jordan is the one exception to this rule as Palestinians can freely travel there without a visa.
As for the actual topic, I would suspect RJ would be the most likely airline as they already serve TLV. TK is an option but Turkey and Saudi Arabia aren't exactly on great terms and for that matter Israel and Turkey aren't doing so well either. Jordan seems to be well liked enough to where they could be the intermediary for something like this. I would think RJ could operate some hajj charters if it gained permission.