Yeah…if Balrogs are never shown flying and fall off things (as happened with Gandalf at the bridge of Khazad-dûm) then it is hard to think they have wings.
Or, maybe, they’re just enormous fiery demonic penguins, with stubby wings. 
My own personal theory (for what its worth, which is probably not even a tharni), is that some elements of the story, like Tom Bombadil, are inclusions from other myth cycles. Tolkien was creating a synthetic epic myth cycle, akin to the Norse Eddas, the Celtic Mabinogion, or the Finnish Kalevala. But real world myths grow organically, and are formed from syncretic mergers of different myths, and include bits and pieces from older myth cycles, or intrusions from later myth cycles, or later cultural encounters with other myth cycles.
In that reading, I think Tom Bombadil and Goldberry and Old Man Willow are inclusions from a different, smaller scale folklore tradition, and the two mythic traditions don’t quite mesh. Hence Tom’s indifference to the One Ring. I think Hobbits would also be part of that same folklore tradition, but they’ve been more tightly incorporated into the Silmarillion cycle, but still incompletely - they just kind of show up in the story part way through.
By that reading, the Balrogs may also originally be from a different myth cycle, and have only been partially integrated into the Silmarillion cycle, where they’ve been syncretized as a kind of Maiar, but don’t quite fit.
Or not.
Even though it has nothing to do with Balrogs, I like to share this whenever Tolkien pops up. Not my work: Oldest and Fatherless: The Terrible Secret of Tom Bombadil - Loose Connections — LiveJournal
My take is that balrogs do absolutely literally have wings, but that those wings are literally composed of shadow. And while wings of shadow might be useful for a spirit creature, they’re not so useful for a corporeal embodiment, beyond spreading them out to look more intimidating. So they can still fall to their (corporeal) deaths.
And if one absolutely must fit in Bombadil with the other parts of the legendarium, I favor the interpretation that he’s the fae embodiment of the World as a whole (as compared, say, to Ungoliant, the embodiment of Darkness, or Goldberry’s father, the embodiment of one particular river). But that’s only if I absolutely must fit him in. The explanation I favor is that he’s just a member of the category of “beings who are Tom Bombadil”, and that there’s nobody else qualitatively like him.
Except it is more properly spelled 've.
Says who? And why do you believe them?
I’ve been thinking about starting a grammar thread. Might be time.
Only when writing ex cathedra (which I am not here). Besides which I hate picking grammar nits.
Ex Caecilius? (Or did we stop doing Cecil jokes?)
One does not simply stop making Cecil jokes…
Would he still have fingers if he was pure energy?
Yes, He has only four on the Black Hand, but they are enough,’ said Gollum shuddering.
Balrogs lurked still, awaiting ever the return of their Lord; and now swiftly they arose, and passing over Hithlum they came to Lammoth as a tempest of fire.
Silmarillion
Now, if they were not literally flying, and it was used metaphorically, that would read “through Hithlum”, surely?
Have you never heard the phrase, “over hill and dale”?
It does not mean to fly over those things.
Sure. Why not? We know he had an eye (or something like an eye).
I’m very familiar with it, since it comes from my favourite play. In the original context, it very much did mean that, given it was a fairy saying it.
We have very different ideas of what “pure energy” means.
Not really interested in nitpickery. Pure whatever demigods are made of.
A lot of people seem to use it when riding and not flying:
Sure, now. It’s become a figure of speech, but was, AFAICT, original to Shakespeare and fairies who clearly were capable of flying (not necessarily with wings - possibly more like airy spirits) given some of what they’re described as doing.
And that doesn’t address the non-existence of an equivalent “over < Place >” figure of speech. Or the overall arose->passed over->rained down arcing motion in the passage.
Hmm, every year i read that the angel of death passed over the homes that had the blood of the sacrifice on their portal. We literally refer to the holiday as “Passover”. I’ve never envisioned that as the angel flying above those homes, just skipping them.
I realize we all have different cultural touchstones, but “passed over” just means “skipped”, or “didn’t attack” to me.
In the animated film “Moses, Prince of Egypt” (1998), the angel of death is portrayed as an amorphous river of pale light that came down from the sky and then oozed through the streets and into the front doors of the unmarked houses.