OK, I can confirm that the Wii Video Mode Changer seems to make no difference with Brawl.
I have my Wii setup in 480P on my 42" LCD, and when launching Brawl from the disc menu it used to flick to 480i mode. This still occurs and the game plays as normal.
When I set my Wii to 576i mode, Brawl launches, but I get a black screen for a bit, and all of sudden it comes back but there is some weird graphical glitches happening. It also seems to slow the console down, all the menus feel very weird. Jumping back to the Wii Menu, the slowness and graphical glitches are still there. I had to power off my console and power it back up before things went back to normal.
Set to 480i mode, everything just works, but I'm pretty sure this is the case even without patching.
For reference, I patched using the Smart Mode and v2.1 of the Wii Video Mode Changer. Here's a quick summary:
Wii set to 480P --> Game switches to 480i on boot
Wii set to 480i --> Game plays as per normal
Wii set to 576i --> Game runs but very weird graphical glitches and slowness.
Hope this helps. Guess I will have to wait a bit longer for a 480P version...