Even in Classical Mechanics (CM) the variables in the Lagrangian are unknown functions, not the solutions to a particular problem.

In QM, apart from equations for unknown variables, there are some commutation relationships; the latter make the solutions of particular problems "quantized".

Normally, the equations of motion are guessed first, and only then Lagrangians are constructed to "derive" them ;-)

Even in Classical Mechanics (CM) the variables in the Lagrangian are unknown functions, not the solutions to a particular problem.

In QM, apart from equations for unknown variables, there are some commutation relationships; the latter make the solutions of particular problems "quantized".

Normally, the new equations of motion are guessed first, and only then Lagrangians are constructed to "derive" them ;-) No wonder Lagrangians contain the corresponding variables to be found.