Best travel itinerary to Cairo?

That sounds about right. We had our layover in Rome from Chicago on our way to Cairo.
None of the major airports in Portugal have flights to Cairo however.

It might make sense to check flights with one or 2 plane changes and stay in one of those.
I use Matrix Air which is the old travel agent flight system that went public. It doesn't sell flights and has more airpine choices than say Expedia. Put in LA to Cairo, flights come up. At the top, you can choose the flights with one layover or two. Look at what is a reasonable cost and arrival time it Cairo (and perhaps a layover city of interest) then price or book as a multi city.
It can take 3 hours to get to hotel from Cairo airport, so keep that in mind.

In mid-March, Portugal might be too chilly. I'd choose Barcelona over Madrid. And although it's farther east, I'd consider Istanbul because Turkish Air is a better than average airline and is usually cheaper too. You will have to get visas for Turkey - I think it's easy to do on-line and not expensive.

Not sure what airline miles programs you are a part of, but we prefer to fly Delta and have found that flying into CDG Paris or Amsterdam are two options that work out well with partner airlines.

Flying into Lisbon or Madrid direct from the USA is more difficult to find as well as to Rome.

Barcelona may have more options, you have to use matrix or Kayak to check out flights to compare prices and itineraries.

I have been to Egypt twice and lived in Saudi Arabia for five years in the 80s. Egypt is loaded with ancient history and a great trip, just be careful, don't go off on your own, drink the local water or any food not in at least a 4 or 5 star hotel restaurant.

Going on a tour is best as you probably found.

A great trick I learned on this Forum: to find who flies where from each airport, look at the airport's Wikipedia page. Here's LAX: https://en.wikipedia.org/wiki/Los_Angeles_International_Airport#Passenger

Next, do look at what's involved in getting from LAX to your "middle country" to Cairo and back. Plug your dates and cities into Kayak or Google flights, looking at getting the flights as "multiple destinations" or "multi city" all on one ticket.

Thanks to airline alliances, you'll have lots of options. And since there are no nonstops from LAX to Cairo, you'd have to change planes anyway, so I like your idea of a stopover of a few days. However, I'm not sure how much shorter the flights actually are to some places. Due to curvature of the earth and the actual routes flown, sometimes "closer" places aren't that much faster to get to. For instance, New York to Tokyo is about 14 hours, and New York to Seattle is 5 hours. But, San Francisco isn't 9 hours (14 minus 5); it's 11 hours.

If you find that the times are going to be long no matter what, look at using Turkish Air with a stop in Istanbul. Istanbul is a fascinating city, Turkish Air is a nice airline (far nicer in coach than others) and there should be frequent connections from Istanbul to Cairo (other cities probably have more limited options, but I haven't researched it directly).