We have always used car seats on airplanes for our kids until they are 4 or so.
I am not a big fan of flying so, I feel much better knowing they are safe and sound in a car seat.
We have never had a problem bringing them through the airport. We had a stroller in September for our DD so, we put the car seat right in the stroller and wheeled them both to the gate. They took the stroller and we took the seat. The flight home was really late so my daughter sat in her stroller. I just loosened the straps of the seat and wore it like a backpack or on my arm.
The seat also helps with keeping my daughter still. She knows she can't get out of her seat in the car so, the only time she asked to get down was to use the restroom. She just sat in her seat and colored or watched the portable DVD player. There was two small kids in front of us that were not in car seats that spent the entire two+ hour flight looking over our seat and the one in front of them. We didn't mind but, the parents spent the entire flight telling them to sit down and apologizing to everyone around them. My husband and I felt so bad for them by the end of the flight.
So, my suggestion would be to bring a car seat for your 3 year old and a portable DVD player.

Heck, my husband and I are going down for three days next week and I am bringing our DVD player for me, lol!
Have a great trip and safe flight!!