I actually think bird poop, if cleaned daily, has almost no smell--if left for a long time to pile up in little poop mountains, perhaps you might start getting a bit of a "chicken smell," but of all my birds the only one whose poop has an actual noticeable odor is my dove. My parrots' poop has never registered to my nose as an unpleasant smell.
So I would look elsewhere in the cage. Do you dump & scrub water bowls daily? Those can really build up bacteria quickly, even if your bird isn't making "cereal" (lol
@PeentPeent my birds do the same, but we call it soup

) they're still eating and then getting crumbs from their meal in the water when they take a sip. If you feed veggies or fresh chop, then any food flung from the bowl onto cage bars, perches, or the wall will eventually start to smell, so you need to stay on top of cleaning that. This may be difficult if your bird isn't trusting enough to leave the cage, as you'll most likely need to use a cleaning tool to get caked-on food waste or poop off of perches, and that will almost definitely scare your bird.
If your cleaning regimen is stellar and your wife still complains of smell, it's honestly probably bird dust. Birds are extremely dusty animals, and if they don't regularly bathe themselves they do get kind of a musty smell... idk if musty is the right word, but it's definitely a scent lol. Typically, birds lubricate their feathers either with oil from their preen gland or with dust, or a combination of the two, and according to Google, greys are the dusty type. Some dust is normal and healthy, like I said it's how they lubricate their feathers and it also kind of waterproofs them, but if they never bathe then it builds up and has A Smell(tm). If your wife describes it as being kind of a sharp musty odor, subtle but kind of sticks to the back of your nose, that's how I would describe bird dust smell, although my parrots have to really get in my face for it to bother me.
Ugh, sorry, I always write walls of text, I really hope this is helpful. Anyway if it's a dust issue you just want to encourage bathing. Providing a water bowl big enough to hop into in the cage is a start, he may start flicking the water on himself the moment he has a large enough water source to fit. Some birds don't dunk though, some birds prefer running water or spray bottles, so you can try shower perches or letting him near a gentle stream of water from the faucet. Some birds like cold water, some like lukewarm, it can be a bit of a struggle to figure out the perfect conditions to convince your bird to bathe, but the key is that if he's ever uncomfortable with getting wet, don't push the issue because you're going to end up making him hate water even more if you force it.