+1

since Steam apparently doesn't let publishers do free key giveaways anymore without also doing a free promotion on the Steam Store.

Have you seen any clear indication of that being the case?
Thus far, the information I've seen has suggested the restriction only applies to storefront discounts, subscriptions, and bundles, wherein equitable sale offers must be offered on Steam "within a reasonable amount of time" if any other storefront receives a discount higher than what Steam has received.

From the paragraph I see quoted elsewhere (below), there doesn't yet seem to be any clear indication that free promotions are covered in the matter, as they are not technically disadvantageous to Steam consumers (as anyone can pick up a free key, it isn't disadvantageous to anyone), they're not a discount persay, and they're not exclusively mentioned anywhere in the documentation for the matter nor properly fitted to the scope of what the documentation does explicitly cover.
Moreover, as a more notable technicality, free promotions don't have to be run through a storefront or even a storefront's standard purchasing system- as such, so long as the latter isn't used, the discount isn't officially being handled by the store in question, thus evading the guidelines as-stated (as then it would be discounted on a site, rather than on a store). Beyond the second part, the first part means that any non-store site would still be permissible regardless.

Again, if you've seen information to the contrary, that'd be another thing- but thus far, just off what I've seen mentioned, the intent of the new guidelines appears to be targeted exclusively at handicapping bundles and other retailers, not at targeting things like free promotions, direct-from-developer purchases, or backer sites. Put more simply, it gives the impression that it's driven by Valve's greed and attempt to weaken competition, and as offering a game for free a second time on Steam would perhaps only cut into Valve's potential profits, it may not really fall under the purview of what they're trying to work against.

It's OK to run a discount on different stores at different times as long as you plan to give a comparable offer to Steam customers within a reasonable amount of time.

Occasionally it may make sense to offer your game in a bundle or subscription, timed at the right point in a game's life cycle. Keep in mind that the perceived price in the bundle/subscription should be a price you are willing to run the game at a standalone price or discount on Steam. For example, 10 games in a bundle that costs $5 USD may have a perceived price of $.50 per game. Similarly, 10 games for $10 through a subscription this month may have a perceived price of $1 each.

tl;dr version:
There's no indication at all in the official documentation (that I've seen) that free games are covered by the new guidelines, and the documentation instead actually seems to indicate the opposite.

Happy to help! 😁
One more thing, I forgot to mention that if the Steam sub is not for sale it won't show up if you follow the above link, in which case you just use StemDB, i.e. steamdb.info/sub/8962/apps/