Does stainless steel affect garlic flavors in food?
Drawing from some of the linked resources, it seems that one of the main theories about how those stainless steel "soaps" work is by interacting with sulfur compounds present in onions and garlic (together part of the genus allium) which are responsible for their strong pungency and odor.  There appears to be little actual evidence for these claims, and I wasn't able to find much better either.

That said, I'm going to surmise that this effect really wouldn't apply with stainless steel cookware, precisely because you're cooking.  Those sulfur compounds are volatile, and they tend to break down pretty quickly as you cook.  This is why the flavor profile of alliums changes so drastically even after a very brief sweat - they become much sweeter and much less sharp - because those sulfur compounds are being broken down and driven off.  Heck, onions will lose a lot of their harsh pungency just after being chopped.  Dice an onion and you'll likely tear up (due to the action of syn-propanethial-S-oxide gas) but come back to that same onion 3-4 minutes later and you'll experience much less irritation.  The compounds are generally unstable, especially when exposed to enzymes present in the onion.

So, cooking alliums in general removes or destroys the same pungency that causes the unpleasant odors.  Even if stainless steel interacts with the compounds responsible for that pungency in some special way, it's only doing what heat does anyway, and using stainless cookware is unlikely to cause any sort of perceivable difference.

The only time this might be a factor in the flavor of a finished dish is if you want to retain a very strong, pungent flavor, for example with raw spring onions in a salad or something.  In that case, it may be possible (though again, evidence is mixed) that a stainless steel bowl attracts some of those sulfur compounds in a way that a plastic or wood bowl does not.  But that's only a concern where you have contact between the alliums and the bowl, and many people find that pungent "bite" a bit overwhelming in any more than tiny concentrations, so you may actually be doing yourself a favor and providing some minute safeguard against over-pungency by using stainless.  In my opinion, this isn't likely to be a major concern for anyone.