[Keeper of Dol Guldur]

To risk sounding like philosophy class . . .

If by 'flame of anor', Gandalf meant the ring of fire's fire was sunlight, and if the Balrog, as a creature from the beginnings of Melkor's foul schemes, the Balrog like other creatures could not withstand the sun, then I don't think the Balrog could have used the ring of fire.

Like someone mentioned, the Balrog kinda stuck to hanging out in Moria, and he wouldn't likely know anything about the context of the Ring of Power . . . he'd know it was an elvish tool of some sort, sure, and know it had the power of the sun, but it's not likely he'd know it went along with Sauron's schemes and the One Ring (if he knew of the One Ring at all).

The elvish rings weren't meant to be used as weapons, and every time we've seen 'The Wise' use any sort of magic, it's been basically with the three rings used in conjunction with other sorts of instruments. Gandalf's white rays of light seemed to come from his hands later on, but he used his staff for the most part in the beginning . . . possibly channeling the power of the ring through it. The same goes for Galadriel's mirror and the Phial with her ring, and Elrond's control over the floodwaters of Bruinen.

So anyway, if Gandalf broke his staff, and the result was spectacular white light surrounding him, it seems maybe that staff wasn't quite capable of releasing the necessary power to dispel darkness. Possibly a reflection on Gandalf himself, he was old and tired, and worn - like the staff. Later on as Gandalf the white, he certainly seemed more sprightly and vigorous, and capable of blasting off a couple of channeled white rays at the Nazgul.

Anyway, that's all I've got to add for now until I've reread and rechecked a whole lot of stuff which bears importance to this topic.