Gail is right, a lap belt won't hold a baby. The FA's know it. You will be told to hold the child during takeoff and landing, and also during any periods of turbulence there might be. Therein lies the danger. In the event of turbulence, if the plane drops suddenly, a child held on your lap will shoot straight up; slamming his head into the overhead bin. I know you would hold on as tightly as ever you could, but against those sort of g-forces; your grip is unlikely to be strong enough.
Most of my family live outside the US. My son's current convertible booster carseat has logged over 80K miles in the air, and the toddler seat before it traveled twice that far. He does not use his carseat in planes now, as he has finally passed 40 lbs., however, it still flies with us as a checked bag, in a satchel with backpack straps. I'll be having another child in December, so I'm about to tackle the challenge of checking one carseat while carrying another one onboard. You CAN do it; it just takes careful organization. That's par for the course when travelling with a baby, anyway; as long as you have to learn to manage a system, you may as well opt for safety and personal comfort on the flight.
Just as an example of one way to handle the process, I'll share my system. (My apologies to anyone who has heard me outline this system before, but since the search function is down so often right now, I'm going to go ahead and repeat myself.) We tend to jokingly call this the "hand rule," and it goes like this:
Count the number of people in the party capable of handling luggage. Multiply that by 2 (presuming that each person has 2 good hands to carry bags; if not, count accordingly.), then, subtract 1 for each young child in the party. The result is the number of bags you can comfortably take to the gate. You also get to count one stroller as a freebie if the child(ren) are riding in it through the airport. With two healthy adults and one baby, you can take a stroller, a carseat bundle, and a maximum of 2 carryon-sized bags. No more than 2 of the items can be rolling pull-behinds, as the person pushing the stroller cannot simultaneously pull rolling bags. If you have small items like Mom's handbag and a camera bag, consolidate them into one larger carryon; you do not want to be juggling small items. At security, send one adult through, then when that person clears, have the other adult go through carrying the child. That way, one adult can keep an eye on the child and the luggage at all times, even if one is pulled out for a search. The stroller and the carseat may have to be hand-searched; be prepared for that possibility. We carried the carseat in a backpack satchel, which left hands free to push a stroller; backpacks are going to be your friends!
To backtrack a bit to the preliminaries: if you drive to the airport, drop one person at the curb with the baby, the stroller, and the bags to be checked, while the other adult goes to park the car, (taking charge of the carseat and the other carryons.) If the child has a purchased seat, the adult who has been dropped off should be able to check the bags with the aid of a skycap. If a trip to the ticket counter is required, the adult and child get in line w/ the itinerary, and tell the agent that the spouse is coming shortly. The agent will process all of the tickets, and print the boarding pass for the absent adult, but will not hand over that boarding pass. However, when the missing adult arrives, he/she can bypass the line, and need only show ID to the agent to claim the boarding pass.
Others here have already covered how to handle the bags on the way out: either tip a skycap to help with the bags, or spring for a SmartCart. I recommend restricting yourselves to 2 checked suitcases for the whole party if at all possible; you should be able to handle that many without help, if you bring a bungy and strap the carryons onto the checked bags for the walk to the car. If the luggage you have now are all 22"rolling carryons, I suggest investing in a good rolling pullman that will meet current size and weight requirements. Those 22" bags are great for business travel, but you are better off consolidating into a larger checked bag when travelling with a baby. We take a rolling hardside ice chest as luggage to WDW (milk storage is important), so our one other checked bag is a 29" rolling pullman.
PS: I forgot to mention that toddlers tend to get very excited on airplanes. Expect the baby to stay awake until about the last 30 minutes of the flight, and then not want to wake up when you disembark.
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