[Galadriel55]

Moreover, Gandalf points his staff at the things he wants to set on fire.
And where did he get a new staff from after he broke his on the bridge of Moria?

I think that the magic that anyone does comes from the willpower of that person. Incantations and staffs are probably just used to help direct the power to the correct action. However, people with inner power don't need staffs. If you define magician, you'd get someone with supernatural abilities. Galadriel does "magic" when she works her mirror. And she says that she can make the nirror show whatever she wants with her will. Aragorn enables the Grey Company to make the trip from the Path of the Dead through entire Gondor without much rest with his will, as is noted on a few occasions.
This would mean that wizards will still have their power if their staffs are broken, but it would be harder for the to release it to specific things.

By the way, I like the analogy with the policemen.