I just learned about the Connect feature by chance and tried to use it right away. The account is connected, but there should be 3 games I can activate on GOG right now. I have set the Steam profile to public (only the top-most option, not the others), but Connect doesn't find anything. :(

And since this feature has been around for 1 or 2 years already, I'm wondering if supported games return every now and then and how often? Or is it only a few titles that keep coming back? Only new(-ish) games, few old ones?

Connect games typically don't return. The feature is barely used, only during big sales to boost attention from customers. It can take some time to work its magic. Don't wait, write a detailed letter to support.

EDIT:
https://www.gog.com/forum/general/gog_connect_does_not_connect/post22

How would you know what games to start if you don't know what games are available to you from GOG connect?

I don't know. The OP knows what games are available. If you need to know, someone can post a list or screenshot of the connect games available. But if you can't see the list, I don't think your accounts are properly linked and therefore you won't be able to activate your Steam games on GOG.

This whole thing is crazy. Every time I think I learn something about the process, I find out it's wrong. I have already redeemed a few games through GOG Connect, but I didn't even have to make my Steam Profile public to do so. I thought GOG had to notify you if there were new games available on GOG Connect, but there is nothing showing right now on the website, so I'm wondering how he's getting games now. They also seem to expire quickly.

If your profile isn't public then you won't be properly linked. You have 4 days left on the current batch I believe. I can't see since I have no more eligible games.

Good luck. If you have problems, contact support. I know I added 2 games from my humble library to Steam the other day just to pick up the titles here (Sexy Brutale and Orwell).

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Since I don't have a Steam account linked, I am -of course- not eligible to any of them, but I can still see which ones are available. Is it different if you do have an account linked?

Currently available games:
AI War Collection
Hunie Pop
Kim
Mount & Blade
Orwell
Pirates! Gold Plus
Tempest
The Sexy Brutale
Torchlight
Tropico 4
Unreal Tournament GOTY

The list already tells me which ones I do and don't have on my GOG account (the title is in grey for owned and black for not owned, not very prominent, but the difference is there), and thus I know which ones I don't have:
AI War Collection
HuniePop
Orwell
Tempest
The Sexy Brutale

So, if I had those on Steam, had my account connected, and GOG hadn't yet noticed some of those as eligible, they are the ones I would launch on Steam.

Thanks for the helpful information. In regards to your question, the only thing I have ever seen is an option on the front page to connect to GOG Connect. I assume everyone sees this. When I click on it, it's supposed to show me if there are any games available. I have never seen any list of games that are currently available. I always wondered why they didn't show that. Where did you get your list? Does it show any expiration dates? At one point GOG said Torchlight II was available through GOG Connect, but I could never get it to work before it expired.

UPDATE:

Wow. This is embarrassing to admit, but I have never bothered to scroll down the page to see the list of games. I didn't even realize they were there. All I saw was the refresh button and the message that I have no new games available. I might not have noticed because I have the font size set to larger than normal. I guess I see the same list you are talking about now.

Thanks. I just figured this out!