It was your call to make as DM, and you have valid grounds on which to make it
There is a RAW reason this doesn't necessarily work on a stirge: it says that it works via
us[ing] your wit to distract, confuse, and otherwise sap the confidence and competence of others.
As I imagine you judged, a stirge is pretty impervious to distracting or confusing witty words, being of low intelligence and having no linguistic capacity to understand any cutting words. There is no hint of magic here to transcend those barriers, either. (The whole description of the College of Lore makes itself out to be about the clever application of knowledge — emphasis on cleverness — not magic, leaving an impression that any ability that's not explicitly stated as magical in reach, simply isn't.)
In D&D 5e, there are many effects that punt to the DM to determine their applicability, based on the DM's in-the-field judgements about the nature of the fictional reality they're collaborating to create with the players. The game does that because it defines world-creation as the DM's job. (Check out the DMG chapter on planes for an extreme example of this.) D&D 5e is riddled with little details like this that are left up to your judgement, all the better to make the rules fit the world you're running, rather than forcing your world to adjust to nailed-down rules. And in D&D 5e, there is no such thing as a fluff/rules divide that anyone can point to outside of their own personal judgement — and it's the DM actually running the game whose judgement matters, not my or someone else's opinion on how these details should be decided.
So, you made the right judgement — for your game. Bards can't befuddle stirges with cutting remarks about how their carapace is so last season, ugh, or try to distract them with clever patter — in your game.
That said, the fictional reality of some other DM's game could easily work differently. The point is that your on-the-spot ruling was entirely justified, because it reflects the reality of your game world, and representing the game world to your players is one of the DM's most important roles. Another DM would be equally justified to say that even bardic tricks that don't mention magical underpinnings are inherently magical; this judgement would portray a subtly different kind of world, and grounding that portrayal in functional abilities is the most effective way of making the world real to the players. Equally, a DM could rule that the bard has particular knowledge of body language (or other mundane communication methods) that would allow them to befuddle a particular dumb beast — an example would be understanding canine psychology enough to shake an attacking dog's confidence momentarily. If a DM felt that such a detail of stirge psychology could be known by the bard and exploited successfully with their humanoid body language and vocal equipment, then that DM could rule Cutting Words applicable.
The point is that you didn't make one of those alternative judgements, and your player's objection doesn't override the sole responsibility and power 5e gives DMs to decide these corner cases.
You're the DM in this case, so you get to define the world. In your world, stirges aren't susceptible to witty remarks. Your judgement is supported by the game's text: it does not say or imply that it's a magical effect that transcends all comprehension barriers; the description supports an interpretation that the fictional actions involved could simply be inapplicable; and because RAW explicitly gives you, the DM, final say in such corner cases.
A final note on fluff and crunch
There really is no way to separate fluff and crunch in 5e. The fluff/crunch divide is wholly made up by fans, and nowhere in the rules as written can you find an instruction that says "these things are just flavour and can be ignored, but these things are rules." Nowhere — and anyone citing "fluff" as a reason to ignore actual written words in the rulesbooks is importing a convention from some other game that isn't D&D 5e.
Look at it this way: if the action was more obviously physical in nature and less kinda-social, nobody would be calling it "fluff" that can be ignored. Imagine a Swashbuckler subclass that has this ability:
A swashbuckler can use their training and agility to swing from overhead features of the environment. When a creature that you can see within 60 feet of you makes an attack roll, an ability check, or a damage roll, you can use your reaction to expend on of your uses of Swashbuckler Flair to knock them prone. You land in a square of your choosing beside the creature.
Nobody would be arguing that "overhead features is just fluff, sure you can use this on a featureless prairie". Ditto for Cutting Words: the description of the ability is rules, just like every other word in its description. And it's up to the DM at the table to make rulings on the significance of that description-as-rules to the specific circumstances that come up in play.