Lots and lots of games supported the six button. Mostly fighting games... I know Comix Zone supports it
And yes, the mode button has a purpose... if you hold it down while turning the machine on, the six button functions like a normal three button. This was necessary for some games, Ristar comes to mind. It would randomly lock up if you used a six button on it.
i have super street fighter 2 and it supports it. i probably have a few other games too, but thats the only one that comes to mind (besides wing commander for sega cd, which is barely playable without a six button)
Streets Of Rage 3 also supports 6 buttons, let you do their finshing combo moves stright off (like Axel double kick) and back attacks which normaly have to be a different way (Holding B button for a while then letting go *e.g Axel Double Kick*, or pressing B, C buttons at once *Back attack*)