No no no, you've got it all wrong. Your cats look more like this.
More seriously I think there's a disconnect between you and the mechanics. They're focused, perhaps by past jobs, about the odor getting in the cabin. Replacing what are perhaps damaged and leaking exhaust system parts are a likely cure to that. You are concerned about the actual exhaust odor no matter where it's at.
This is a bit of a grey area because cold starts, for emissions testing when new, is at 68F, or not cold at all. They're probably also looking at it with a view on possibly passing smog, which they interpret as meaning the system is operating as well as could be expected, which often these days means no OBDII light on. But it's not like it once was? There's basic tune up like spark plugs, more in depth like O2 sensors, the front ones, which are a b***** to replace, but not a whole lot more unless there is some sort of an air pump that runs during cold starts, which I'm not aware of, but that's not saying much. Rear O2 sensors are a complete red herring, they are monitoring only and have no role at all in emissions, they just tell you if there might be a problem. Which leaves the cats, no matter what they look like. A high quality cat will last almost forever under good conditions. Good ones last 4-8 years. Cheap ones a year or two. But even cheap new cats test crazy clean and good or very good ones do that longer.
All of which is a long way of saying if the check engine light is off, change the plugs and O2 sensors, the front ones. If that doesn't so it you're looking at cats but if you're planning on keeping it buy OE or the stupid expensive California legal ones, they should last longer. Not that cheap ones won't fix it. Just not for long.