I have heard the same.
Anecdotally, I believe that I read somewhere that you could ask Israeli customs officials not to stamp your visa because you were going to (or may in the future) be travelling to a Muslim country.
I could easily picture a Syrian customs official denying entry to a normal US citizen who had an Israeli visa stamp on their passport - but seriously would/could that same customs official actually deny entry to someone who was a State Department diplomat?
That is true..Israel’s been doing that for decades..
Two decades ago while working in Yemen, we were warned not to get an Israel stamp in our passport if we wanted to return to work.
We were also told by leaving a blank, loose piece of paper in your passport and explaining where you were working BEFORE handing over your passport, Israel customs would stamp the loose paper in place of your passport.
It used to be that Israeli immigration would give you an insert to stick in your passport if an Israeli visa stamp might prove inconvenient. A number of U.S. defense contractor personnel are Jewish and visit Israel and Saudi Arabia regularly on business. Most of the Saudis do not mind doing business with Jewish Americans, they feel they are getting the best that money can buy. The Saudis actually have a great deal of respect for the Israelis, and appreciate that they are keeping Iran off their backs, for now. The whole antisemitism shtick is boob-bait for the Arab bubbas.