Hitler could not send an Asien Corps, or at least successfully, sending troops to North Africa is a short hop from Italy to the coast covered by the Italian Navy and land based air. Syria, much less aircover and farther from the Italian Navy, which was optimized for short range ops, bases, they could not protect such a force from the Royal Navy
US public opinion was quite willing to go to war with Japan according to polls from the time. Anyways once a US/Japan war starts and Japan takes a significant number of US POWs it will be a war to the finish, given the way the WWII Japanese treated POWs
Sending some forces to North Africa to keep the fighting away from Italy longer was a good decision, keep the Italians in the war longer. Sending OTLs amount of forces and attempting to actually conquer Egypt was stupid
Logistics, the Axis can't push enough supplies to the front to beat the UK in Egypt, can't really access Syria (further away than North Africa, RN in the way), Vichy lacks the stockpiles to really undertake offensive operations in West Africa (and are likely to go Free French if ordered)
Rabble rousing among the Arabs is fine, but logistics for anything much more than OTL are tough, distraction, nothing more
To really open up more fronts need a POD years before the war