I’m positive you are all kidding rekindling this old debate again (I’m sure Michael Martinez, for instance, has felt a shiver going up his spine these days)
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But for the sake of it... and considering more than anything that The Silmarillion’s & The Fall of Gondolin’s Balrogs have very little to do with the Balrog of the Lord of the Rings... I daresay that LOTR Balrog has also nothing to do with Peter Jackson’s films’ Balrog... I think that Professor Tolkien depicted it (him) not as a horned (and winged) volcanic monster, but as a dark human (but bigger in size) shape shrouded in shadows and (at will) flames. That aura of shadows (and flames) he could expand or shrink, causing a very similar visual effect to real wings for the (terrified) observer.
Could the Balrog of Moria fly or not? Not sure about it, though I think I recall the text uses the expression “to leap over the chasm” and not “to fly over the chasm” or something similar... and nothing in the text suggests that the Balrog tried to (or could) fly up while burdened with Gandalf’s weight... he fell like a stone to the lake at the bottom of the Chasm.
On the other hand, I would find nothing weird in a flying Balrog, as he was a fire & shadow spirit... and others among Morgoth’s minions could fly too (and before winged dragons came to existence)... Vampires like Thuringwethil, for instance.