The real question is "does it cost more to certify the autopilot to do this than it does to pay for an extra two pilots", and with my understanding of aerospace costs that answer would be "very, very much yes".
I could imagine the autopilot could do a much better job.
In the ideal 'zero-g' flight, you can release a ball at the start, and it should stay stationary (relative to the cabin) for the whole flight.
That means the plane body must be to within a few inches of the exact perfect location in the sky 30 seconds later.
I haven't flown planes, but I imagine that making sure they follow precisely a path with an accuracy of an inch in all three dimensions while travelling at 600 mph is no easy feat. But an autopilot with a fast control loop might be able to do it.
All the autopilots in the world aren't keeping a plane to within an inch in all three dimensions. It'd be nice, but turbulence is a thing :)
More importantly, actual position isn't that important - you care about relative acceleration being as close to 0g as you can.
And with the 3-pilot setup the A310 zero g flights have, they maintain +/- 0.02g. Now you add to that the fact that developing an autopilot you can trust in that situation costs a good chunk of money, the fact that it's a specially modified plane (so it's more or less a one-off effort), and the fact that pilot costs are negligible compared to the rest of the flight cost.
At this point, you can get a minimal increase in precision for a large expenditure up front. You'll still have the 3-pilot setup (you want to be able to recover on malfunction), so operational costs aren't reduced.
At that point, the question becomes "why would you"
The “easy” way to do it is to mandate autopilots to have the capability for precise ballistic trajectories. This would need better positioning, accelerometers, and meteorological radars and would also enable fully autonomous landings (if the plane can be aware of air densities and speeds around it, it can nail the landing even with wind gusts).
Wait what? If your landings are a ballistic trajectory, kindly step away from that stick ;)
And that's kind of the point - nobody except the few planes that fly parabolic trajectories needs this. So good luck mandating something affecting all of aviation with no practical use except for a handful of flights.
Autoland already exists. And can handle crosswinds up to 25kts. It's just that using them is an incredibly intense task - your reaction time to fix mistakes is a bit short.
That's why most landings are still manual, unless visibility is so bad that you have no choice.
(Also, the idea that meteorological radar could give you enough info about air density to preemptively handle air gusts is... a bold future)
I didn’t say parabolic landings. I said the same capabilities required to fully automate parabolic flights would allow safer landings because precise trajectory control even when there is turbulence around the plane is a very desirable feature.
On second thought, it could pay for itself in preventing aborted landings.
Most passenger airliners flying today can land automatically, but are only allowed to do so when visibility is so bad that it's the only option. It's called CAT III Autoland.
There isn't much turbulence, wind gusts etc. at altitude - these are more of an effect of the bumpy and uneven, alternative hot and cold surface at ground level - the bumps are usually all smoothed out by the time you are high up in the air.
(besides large storms, of course)
EDIT: Turns out there is sometimes some turbulence at altitude, caused by large mountains, storms and the jet stream - but these are all predictable and avoidable.
In the ideal 'zero-g' flight, you can release a ball at the start, and it should stay stationary (relative to the cabin) for the whole flight.
That means the plane body must be to within a few inches of the exact perfect location in the sky 30 seconds later.
I haven't flown planes, but I imagine that making sure they follow precisely a path with an accuracy of an inch in all three dimensions while travelling at 600 mph is no easy feat. But an autopilot with a fast control loop might be able to do it.