I'm here to report my comparison using the wiimote, the classic controller, and a wireless gamecube controller on playing downloaded games on the VC. So far I've only done it for Super Mario Bros.
Wiimote (sideways)--works fine, except no analog joystick, "B" key ("1") is a little less convenient. Also, it takes a little more effort to balance the remote (makes your fingers more sore.)
Classic Controller--much better fit. Easier to hold and balance. "B" key convenient. Can use analog joystick for movement, although VERY sensitive (easy to overstep.) Certainly better than Wiimote (but tethered to that Controller.)
Gamecube Controller--easiest to grasp, since it has grips allowing bottom fingers to do the holding. "B" key convenient. Analog joystick works the best since it is not too sensitive (note this is a third-party wireless controller, a Pelican, so I don't know if a Wavebird is as sensitive as Classic.) Best since not tethered to wiimote (this one was wireless.)
All and all the Gamecube controller is best although the Classic Controller is still good, and the wiimote will do a decent job (although least preferable.) I imagine certain downloaded games (SNES) the Classic is better than the GC and others the GC (N64). What I don't understand is why only the GC can work on Gamecube games (especially since the Classic has all the buttons, plus a few more.) They'd sell more Classic controllers (since there's not 3rd party carrier for that.) I believe it may have been that the Classic was released later and they hadn't foreseen the need to sync the wiimote for GC games. I really think they could change this with a simple firmware upgrade. PLEASE?