Why do we feel gravity on a plane?
This is the difference between flying and in orbit. In orbit, you are indeed falling toward the earth, but the spacecraft is too, and you're going fast enough that you keep missing the earth.

In an aircraft, because it's staying aloft due to lift, it is not falling. This is why you experience the pull of gravity on a plane.

Some aircraft are designed to feel weightlessness, see Vomit comet.