Actual Saftey Concerns
Flying without a seatbelt is dangerous, but it is safer than driving, which is why the FAA allows babies without seatbelts- they are afraid you will otherwise drive.
The Chillul Hashem Aspect
Meh. Anyone with kids would do it; just we’re the ones with kids.
Parent’s Sanity
Yes, just put the kid to sleep.
Car Seat: Yes or No?
Yes, very good, if the kid has a seat. Keep in mind that your kid will then be kicking the seat in front of him the entire flight, but that’s just what kids do.