Why should the AOA sensors be placed outside the aircraft? Is it not the angle between the chord line of a wing and a vector representing the relative motion between the aircraft and the surrounding air, in which case, is it not possible to measure the AOA with a simple instrument inside the cockpit, which eliminates malfunction due to exposure?
No, it is not possible to place the sensors inside the aircraft, because the angle-of attack sensors need exposure to the surrounding air, in order to measure, as you mention yourself, "relative motion between the aircraft and the surrounding air".
To my knowledge, no one has invented the method to measure the outside flow from inside. This doesn't mean it can't be done, as one could imagine, for example, an array of sensors using some kind of doppler velocimetry or other techniques to determine velocity and flow direction. It would likely be finicky, and in my experience standard AOA indicators are robust (it blows me away that the 737 has had multiple sensor failures).
There are a few "software only" AOA solutions out there like the one offered by Aspen that do not involve any sensors (aside from the regular pitot system).
Since Aspen's Evolution AOA indicator calculates angle of attack from the flight envelope data received from the air data computer and attitude heading reference system (AHRS) integrated in the Evolution 1000 PFD or Evolution 1000 MFD, and a certified GPS it does not require additional hardware, nor does it require any external aircraft modifications or sensors
And its even FAA certified but as others have noted its only an "estimate".
But to truly measure angle of attack you need information on the fluid through which you are moving
is the angle between a reference line on a body (often the chord line of an airfoil) and the vector representing the relative motion between the body and the fluid through which it is moving
it is true that you need some sort of direct knowledge about the relative airflow to make an accurate measurement of the AOA. But the amount of equipment needed outside the fuselage can be minor
The AOA logic is internal to the fuselage, and no moving parts (e.g. wind vane) are mounted outside.
however this, like any other Pitot tube will have known issues with icing.