Even though I own Democracy 3 on Steam, it shows up as unavailable to redeem in GOG Connect.

Another user posted about the same problem, but it was in the Democracy 3 subforum, so I thought I'd bring it some more attention in the general forum.

Try the solutions posted in the article below:
https://support.gog.com/hc/en-us/articles/115003586109

In addition, try clicking the "refresh" button in the Connect page (which is below your Steam and GOG accounts).

Nothing in the support article helped.

Clicking "refresh" on the Connect page fixed it and allowed the adding of Democracy 3.

But this is a bit troubling - I have owned this game since December 14 last year.

Normally a website that syncs with your Steam library will do so automatically once per day. Does GOG Connect not do this? Have I perhaps missed out on GOG copies of games in the past because I didn't know I had to manually refresh every time I acquired Steam games (or every time I used GOG Connect)?

No, GOG Connect does NOT do that. The responsibiity is on you to make sure that you've claimed all possible games that are available at any given time.

Also, GOG Connect is in no way comparable. It is a changing, limited time promotion which you have to claim, not an auto-sync for a regular maintenance process.

I didn't think it automatically added GOG copies of Steam games, only automatically synced your Steam library so it knew what GOG copies were available.

Is this documented anywhere?

Yes, it auto-syncs when you go to the available GOG Connect page (if your Steam account is linked), but sometimes there are glitches, and it's prudent to resync manually if games you own on Steam are not showing up.