Some perspective on long-distance sea trade
I just used my G.Projector global map software to estimate the distance from what appears to be the northern mouth of the Aqaba-Jordan/Dead Sea/Lebanon strait, about where I'd put the homeland core of this TL's version of Phoenicians, first to the Pillars of Hercules--that's about 35 degrees in a great circle. Then south, through the strait to and through the Red Sea, around the corner to the mouth of the Gulf that opens between Arabia and Somalia, then as far past that to the south along the African coast as would add up to 35 degrees.
That only takes me halfway along the southern shore of the Horn of Africa. Switching over to an eastbound route instead just takes one to the mouth of the Persian Gulf. All those turns do make the route add up!
So, the guesses that actually the sea passage need not radically alter ancient trade relations between say Mesopotamia and Egypt seem to be pretty well borne out. It's farther than you might think from the Eastern Med to Sumeria or Persia by sea, even with a convenient double super-Suez provided by nature. To be sure sea trade is so much more efficient than overland that it still might make a big difference. But then, OTL this route was available, provided goods and people took a short overland leg where OTL the Suez canal eventually went. Or actually in ancient times, they often docked on the western shore of the Red Sea, took a short overland leg to the Nile, and then boated down the Nile to seaports at its mouth. Either way, for economic purposes on a typical ancient scale, these overland options sufficed to make the Red Sea and trade both down the African coast and east to the Persian Gulf and even the Indus river mouth pretty viable; I don't see the freedom to just sail straight through to the Med as exactly revolutionizing the situation.
Actually, there might be more of an effect due to the eastern strait letting ships come much closer to the northeastern part of the Fertile Crescent, favoring overland routes from a port in what is now Syria along the southern escarpment of Anatolia to northern Mesopotamia. This too was a live and major overland trade route and it is not clear (since I foolishly forgot to keep p. 1 open in a tab and can't look at the moment at the modified map!) how much the sea part of it we've added would cut off any particularly obnoxious leg of the trip. Damascus I believe lies squarely on this route and it would be good to know if that town, or some other one that more anciently controlled that route, now lies exactly on the coast of the strait.
Turning to the changed land geography-I think Tirion guessed right in the first place, and in ancient days the main thing is that overland passage is now significantly impeded. It isn't for peaceful trading purposes, much. Nor for uncontested migrations of peoples, or unopposed passage of armies. But if a state controlled one shore or the other of one of these straits, it would make sense to fortify it, in the sense of putting up watchtowers and garrisons spotted so they could mutually see each other and the shores in between, and move quickly to oppose any landings they observed between them. Given time and motivation, it might actually make sense to build an actual wall--either Hadrian's Wall, or Antoine's, I forget which, had extensions along the south shore of the sound between Roman Britain and the Picts. The wall slowed down any attempt to remobilize ground forces after a ship landing and gave the Romans defenses behind which they could communicate freely, and shift forces to best fight off the Picts landed on the other side of the wall.
A defense in depth of the western strait seems like something Egypt might invest in early, and despite occasional failures and conquer attempts to dismantle it it would probably evolve, over thousands of years, into something quite Cyclopian and perhaps more impressive, per mile anyway, than the Great Wall of China. A wall along the west shore for last-ditch defense, built quite close to the water; the sound itself guarded at both ends with concentrated naval force in the form of galleys based on both shores near the mouths to deter or stop any attempt to attack by sea; a hinterland occupied on the east shore, on Sinai Island, with a palisaded Pale fed and otherwise supplied and reinforced via the water and a garrison to blunt any attack, no matter how massive, coming overland on the island. To complete the defenses, of course Egypt would seek to conquer the whole island, but trying to erect comparable defenses on its east shore would be a whole other level of daunting, nor could its north shores be completely guarded. When Egypt is strong and ambitious, they'd seek to control the island with mobile forces and oppose landings and uprisings piecemeal, often fighting pitched battles, but absorb all potential advances on the western channel as much as possible.
Sinai Island is of course terrain in which historically in ancient times, dozens of kingdoms typically coexisted; I doubt the Egyptians would often be able to subdue it all into provinces and they wouldn't be able to hold it long when they did; the Jordan Sound just offers too many opportunities for landings over too long a shore to patrol. The Egyptians might do better to foster an ally (such as say Judah) to control that shore as best it can and buffer their own territory to the west of the southern peninsula.
Anyway, despite my characterizing the Jordan sound shores as sieve-like, I do think that sound would be a real impediment to advancing forces, in that the break in pace it enforces gives a vigilant and reasonably strong opponent the chance to hold them at the shore, or even, if they are good with boats, to contest their passage on the water itself. The Assyrians for instance would have to have a navy of sorts, if only of war-barges, to assail Sinai island. I don't doubt they can get a navy, and send over forces that prevail despite the defenders' advantages, especially if Sinai Island is not politically unified. If they recruit some Sound subnation to become their allies in return for not getting massacred, it is pretty much game over for the other Sinaiese, same as OTL, as the traitor tribe gives the Assyrian steamroller safe passage. Still, on the whole the two straits will serve as very natural boundaries--even if the Egyptians do hold on the west shore, they probably won't typically claim much beyond that march, preferring to cultivate allies to do their fighting for them in the east instead. Assyrians, or Babylonians, might manage to more or less subjugate the island and then advance on the western wall, break through, overwhelm the defenses of the west shore wall at some point, and finally advance on Egypt; failing that, they might at least fortify the west shore of Sinai against the Egyptians; then of course no one holds the strait itself, it become a no-mans-sea that peaceful commerce cannot pass through without the unlikely event of both powers permitting it, and yet neither shore is safe from a landing from the other side either. A world-conquering empire like the Persians would probably persevere in breaking into Egypt and subdue the whole place; they might retain the barrier wall system (which insofar as Egyptians built them over thousands of years, might be impossible to tear down completely) but controlling both sides then and only then would the straits be the negligible barriers many have been assuming generally.
Could a Sinaian kingdom arise that holds all the shores of Sinai? I neglected to mention the south shores of the peninsula itself because that is tough terrain and because Egyptians wouldn't worry too much about attacks from the south--maybe they should what with Punt being there and all that. A bold Persian attack might come by sea around Arabia though they'd probably be observed and the alarm sounded. However a Sinaise kingdom would need to worry about the south shore because the Egyptians could attack that way, and to garrison that shore would be a heck of a logistical challenge. Again I think it is most likely if they are in a close relationship with Egypt rather than totally independent.
How much would these two straits slow up Alexander? Conceivably they might divert him from trying to consolidate his hold on the whole Med shore before striking east into Persia; he might, having secured Anatolia and the north of the Levant, simply march southeast along the sound and turn east along the trade route to take Mesopotamia from the north, and leave Sinai and Egypt for another day after he has crushed the very head of the beast. Doing so leaves all the shores of his conquests and even Greece itself exposed to Persian naval raids and possible counterlandings in his rear. If he can move fast enough this may not matter, but having to double back west and first cross over to Sinai, then from there to Egypt, might well divert him permanently from trying to reign in Mesopotamia and from ranging farther east to India, not at any rate beyond what it took to bring the last Persian holdouts to heel.
Or he could proceed as OTL, doggedly securing a foothold on Sinai then subduing it (making up for time lost crossing by having a natural barrier on his east flank to fortify while his main body drives on southwest) then battering his way into Egypt which once past that wall should fall to hum as OTL, then regroup somewhere along the Jordan Sound for the final phase--remember he already holds the north shore at least some distance from the Med, so he could just ferry back up to there and go from there.
I think I've taken this farther into OTL parallels already than many would think reasonable. But I do contend that on the whole, these passages don't make for persistent, systematic butterflies of a world-changing type until at least this era, so close parallels of OTL nations are not unreasonable until nautical technology develops enough that the open sea route makes a big systematic difference. Which is just around the corner at this point.