@Grier
Thank you so much 
Buy a one-connection ticket to start unless you aren’t in a major city.
Tootsie roll lollipops.
Do they still make those inexpensive “umbrella” strollers? Get one of those. They’re light, small, take a lickin’ and keep on tickin’. The parents I see with the big over-engineered strollers, while definitely di luxe, seem to be struggling. Those cheap ones from the 1990s are the ticket. Donate when you’re done. Then let the kiddo run as much as possible during the connections. Locate the kid areas ahead of time. Do some calisthenics with them, too. Those little muscles need to move, or they get restless.
From Alaska, I quip that “We can get anywhere in 3 flights but it takes 3 flights to get anywhere.” By age 3, our son had been to 8 states and 10 countries.
3 is too large for a chest carrier, and while backpacks work, you’ll have other stuff. I’d get an “umbrella stroller” for $20-$30. Gate-check it if you must, and if it gets mangled, buy another.
I’d suggest a duffel bag with stowable shoulder straps (Walmart has a medium/large one for $40) and for walking through the airport, you have 1) the kid in 2) the stroller and 3) the duffel on your back. So you have both hands free. Within the duffel, have a smaller daypack or two of your airplane stuff and his airplane stuff.
More than the stroller, I’d debate about a car seat for the plane. It has to be FAA-approved for use on a plane, and booster seats are, but he’ll sleep better in his own 5-point harness car seat. But that’s a big thing for you to schlep around. With two parents, I’d haul all the stuff and board earlier to install it while Mom ran the kids around the concourse and only board at the last minute.
A ride-on carry-on seems a needless complication, weight, and bother to me.
Bring something for him to chew/suck/swallow during climb-out (and less so during descent) to move his jaws to equalize air pressure in his ears.
Some favorite books and toys. And some new ones.
We’ve never had a problem taking one kid with one parent through many, many airports (only on land crossings if I’m the only adult). But I’m an old white dude. If your son is mixed race or because it’s Egypt and there have been some high-profile custody issues, you might get more scrutiny. There are examples of forms online to be signed by each parent and notarized (like at your bank) you can bring with you. For going to Canada, I downloaded their example (but then we, a white woman and a white man, were waved through with way too many kids of the same age and with different names to possibly all be ours).