Buy the seat.
You can't use any type of restraint, other than FAA approved (car seat) during take-off and landing. You could use the sling during the flight. They also won't allow you to walk through security without taking the baby out and putting the sling through the xray.
In the rare chance that something happened in the air, you would not be able to hold onto your infant, thus they become a human projectile. Not to mention, having the car seat to have your arms free is priceless, and it provides a comfortable and familiar spot for your baby to be.
Good luck!