Autopilot engagement is specified by the manufacturer, and has to do with the specified height loss specifications.
The L1011, for example, the autopilot can be engaged on the ground (CWS only), and moved to COMMAND, once airbourne.
NO height limitations.
However, having said this, those aircrft on the British/Omani register, cannot engage on the ground....for reasons known only to themselves.
Backward thinking, I would suspect.