You are right that current planes are still pretty dumb. It's an artificial restriction though. Certification in this space is very tough for obvious reasons so there are not a lot of companies working on self flying planes because getting solutions in this space past the certification is going to be a long and expensive process. There are a few though and there have already been unmanned cargo flights.
From a technical point of view, flying and landing planes automatically is not that hard. It's a lot easier than self driving cars. And probably also a lot easier than landing a rocket, which is of course something that is not done manually by spacex. Why would they? It's way too complicated for humans and most of the important things require split second decision making far beyond what humans can even do.
The hard part is going to be fixing procedures and practices, which are optimized for humans communicating via 1960s technology. A lot of what pilots do isn't staying ahead of the plane (as they sometimes refer to it) but dealing with a crazy level of communication overhead of people yelling out courses, altitudes, transponder codes, radio frequencies, and weather information to each other over a really poor radio connection, while baby sitting the auto pilot, running down endless checklists, and micro managing it's gazillions sub systems. It's stressful and even very experienced pilots quickly loose the ability to keep up if they don't fly regularly. Because people fundamentally aren't very good at this stuff. They make mistakes. They mishear things. Etc. It's hard but also very much something that could and should benefit massively from automation. Lots more than we have today. All that check listing can be vastly simplified with automation. All the routine sharing of information can be done in text/binary form and be machine readable so the auto pilot can act on it (with or without human confirmation). Etc. But that requires change and the aviation world is extremely change resistant.
The military will likely get there first. The new B21 apparently can fly unmanned. It may still have a crew but it should be able to operate without one.