What is the purpose of having a robotic arm fly a Boeing 737?
The fundamental benefit is that the operator doesn't have to make a lot of adjustments on "legacy" airplanes, they can "simply" install the arm on the existing aircraft that they already have in service.

And this is the main selling point for such a device, they are not marketing it to operators that still have to buy or lease an airliner, but to those that already have one and might want to reduce the number of pilots they have to employ.

Gürkan Çetin notes in the comments that this device is not certified (yet). That's partially the point of such demonstrations, to convince authorities that it can be certified, and to incentivate operators to lobby for their approval.