Sightseeing on way to FCO for evening flight?

There is nothing special to see between central Rome and the airport. It is not really the countryside but the suburbs.
Well maybe Ostia Antica. But your flight leaves when it may be closed

I realize that there is nothing between the hotel and the airport, but would it be logical to combine a day trip from Rome to going to the airport. In other words, if we got picked up at 7:00 a.m. would there be something of interest to see between then and getting dropped off at the airport for a 7:40 p.m. flight? Could we do anything substantive in that time?

You could approach like this : leave your bag at the hotel after checkout (most offer that service) or lock it up in a locker at Rome Termini train station and then take a day trip somewhere.

This site might give you some idea of easy daytrips from Rome:
https://www.neverendingvoyage.com/day-trips-from-rome/

Sounds like you would hire a car for the day and take your luggage with you. One option would be to visit Tivoli which has Hadrian's Villa and the Villa d'Este. It's actually in the opposite direction from the airport, but not very far from central Rome. If you left your hotel by mid-morning, I think you'd be able to visit both sights and still make the 5:30pm flight.

One thing which is important - keep in mind the Rome traffic, getting back to the airport for flight and checking in. Setting a watch or phone alarm would be key. If you can sched the 7:40 PM, that would give you more time in Rome of course.

With that, have you considered choosing one or two neighborhoods to explore?

Perhaps take a browse through these videos from The Roman Guy - Sean narrates most of them. He has been with the company many years.

https://www.youtube.com/user/TheRomanGuyTours

https://www.youtube.com/watch?v=Z3pTI9je2ZQ - a bit ambitious for one day but a good "guide" to choose what you may like to see (You would not be able to do all of it considering your time constraint.)

To make the most of your day - of course you would need to "map out" the attractions or neighborhoods you wish to visit.

Trastevere is very popular - a hip, fun neighborhood, good restaurants and shops https://www.youtube.com/watch?v=v75H2alRIW4

Monti is near the Colosseum - an "artistic" neighborhood https://www.youtube.com/watch?v=g4XCIANUSXc

Jewish Ghetto https://www.youtube.com/watch?v=LHW4qD9Ez5Y (Nonna Betta is a very popular restaurant)

Campo de' Fiori, The Pantheon, The Spanish Steps, The Trevi Fountain, and Piazza Navonna can be seen in half a day.

The Vatican, St. Peter's and Castel sant' Angelo are close in proximity but farther away from the city centre. The Borgo Pio and Prati neighborhoods are close by

Check out the skyline webcams to get a better idea of Piazzas and so forth https://www.skylinewebcams.com/en/webcam/italia/lazio.html

Another vote for Tivoli Gardens and Villa d’Este with a driver service.

Why not just spend that day in Rome?
If you are only there 4 nights that is barely enough time to see just the major sights without jamming your few days too full.

Mhmm, They will be in Rome already.

After I re-read the OP's inquiry, I was confused

my family and I are staying 4 nights at the Waldorf in Rome in late March/early April. I am about to book a flight from FCO-AMS where we are staying for two nights before flying home.

So is the Waldorf in Rome not close to the airport? I did not even know Rome had a Waldorf, lol.

Well, she/he has some good suggestions.