Zero problem at all.

I've brought back wine from overseas four or five times now. Every CBP agent has a different interpretation of the law, but ultimately, I have never paid a single cent of duty, even when bringing back multiple cases (I brought 36 bottles back from South Africa earlier this year).

The law is written ambiguously with regard to personal exemptions. On top of that, the import duty on wine is exceptionally low, so even if you are charged, it's less than $1 per bottle. I've been told by various CBP officers:

-Not subject to duty at all. Don't even bother us by declaring it next time.
-The limit is based on actual alcohol volume, so at 13%, you'd need 7.7 liters of wine (10.2 bottles) to get 1 liter of alcohol, so anything less than that is not dutiable as it falls under the personal exemption
-There is no limit or duty on imported wine, as long as it's for personal use and not commercial (though bringing several cases back may make it harder to convince them that it's personal!). But you should still declare it.
-Probably subject to some duty, but it's such a minimal amount that it's not worth the paperwork hassle, so go on through and have a nice day
-The Feds don't care at all, but CBP is also tasked to enforce state import restrictions, but you're lucky you entered in California where there are no problems bringing it in

As well, about five different CBP web pages all conflict with each other. At least CBSA has a nice webpage where you can put what you want to bring in and it will tell you if you're allowed to and what the duty will be...

I've given up trying to understand what is actually right. All I know is that to be safe, I should always declare anything, I probably won't have to pay anything, and if I do have to pay anything, it'll be a tiny amount.

The Global Entry machines will ask you if you're carrying it, and you should say "yes." (I might personally feel safe answering "no" if I were bringing 1 single 750mL bottle of wine, but due to ambiguity over volume of alcohol vs. total volume of liquid, I wouldn't feel safe recommending answering "no" for anything more than that.) You will probably be sent to secondary inspection. You will possibly have to endure 5-10 minutes of questioning and an inspection. You will probably not have to pay anything.

That's my experience and my recommendation. I have absolutely no qualms about buying outside the US and bringing it back; it's pretty easy and cheap (or free) to do so, and the quality and cost of wine outside of North America usually makes it a MUCH better value than buying here and definitely worth any minor hassle.

Passing through through other countries on the way back to the US, though, can be problematic...the UAE only allows 5 (5!) bottles of wine at all if entering the country (no limit if not entering and only transiting), and some Canadian provinces have extremely high tariffs for importing wine, even in transit. If bringing home wine, always avoid stopovers on the way home and make sure the itinerary qualifies for checking your bags all the way through. But that's a different subject altogether.