I've been on both sides of this debate.
Before I go any further, please make sure the car seat that you might take is FAA approved -- it will state that on the car seat itself and I have been asked by a FA to see it on the car seat.
When my 1st was younger than 2 yrs., I always bought a seat for her, took the car seat and buckled her in. Safety was important enough that I purchased (at 1/2 price) a seat for her. Before she was 2 yrs., she had about 20 flight segments to her name.
However, now that we have 3 kids (5,3,1), its physically impossible to carry on car seats. In the past, I have done different variations mentioned above -- I've taken a sling and the baby fell asleep in the sling on me (which made for a peacful flight!), I've had no car seat/no sling and a lap rider, I've had a car seat for one child and not the others... I've had an 8 month old in her car seat fight the car seat (and this was the "compliant" one) so hard she screamed for 15 minutes and then threw up all over herself, myself, and the car seat at the beginning of a 3 hr. flight. (talk about a car seat rendered useless... good thing I carried it on, huh?) It was the beginning of the "I'm going to cry so hard I'll make myself throw up" stage. Needless to say, I was thankful for an extra sweatshirt for me and change of clothes for her that I had already packed.
Its all what works for you and your situation.
Would I always buy a seat if I could for the baby under 2 and use a car seat ? Sure, its the safest. But not feasible anymore.
Will I fight a child to get into a car seat on a small confined airplane that we are stuck in for 3 hours ? After the vomit experience, absolutely not.
At this point, I'm usually happy to get through a flight with only minor tantrums ... and that's Daddy!
I've also learned that a change of clothes is an essential to be carried on!!!
I don't rely on the rental car car seats, because they are incredibly scary (unsafe and cleanliness wise) to me... I have car seat bags that I use for the car seats (also good to stuff large things in - coats, towels, shoes, etc.).
good luck!!!