Should we use asynchronous inputs in Moore blocks and synchronous in Mealy blocks?
Your conclusions are not true.

The bottom line is all inputs that can cause a state change must meet setup and hold time for the next state to be definitive.

Take your example of the synchronous L pin being fed by the output of a Mealy machine. If the output of the Mealy machine violates the setup or hold time of the L pin. Then there is no guarantee which bits would perform a load and which bits would not perform a load on the particular clock edge.

It is also possible to have the output of a Mealy machine to never violate the setup and hold time. For example, if the inputs to the Mealy machine that affects a particular output is from synchronous sources, then the output is also synchronous (with some additional delay). Of course, there must be enough budget to handle the propagation delays. But that is true here and practically everywhere.