What are the properties of a good general aviation training aircraft?
Cost is certainly a primary factor in choosing smaller planes. It makes training much more accessible to people with less money to spend.

Small planes are also easier to fly for various reasons. They fly slower and respond more quickly than large aircraft, which gives the pilot much more margin for learning. Being more maneuverable can help them escape tricky situations.

These planes are also generally designed to be resistant to and recoverable from common aerodynamic disasters like stalls and spins. See this related question for more information. There are FAA requirements for single-engine planes in FAR 23.221.

Smaller planes generally also have fewer and less complex systems, which makes the necessary training easier to complete.

They can also operate from smaller airports, which gives a student more choices of where to do their training.

Of course, there are always exceptions to these qualities, and large planes can have similar ones, but they hold true in general.