I woke up in the middle of the night in a cold sweat, for I had sensed a disturbance in the force. Now I see that it was the Sith Lord Sampsa up to his usual anti-gold shenanigans. (I respect you too bro <3)
Your 3/2/1 calculation comes out to 12 vs 11. However, the 3/2/1 paradigm very suspiciously uses even numbers. I would argue that something like 2.85/2/1.33 is more accurate albeit less catchy. Failgold and building wealth wouldn't be such popular strong plays if hammers were truly twice as valuable as gold. If we plug in my numbers we instead get 12.54 vs 13.31 in favor of 2

7

. Ignoring cottage growth, I prefer 2/7 over 3/3
I would say you're undervaluing the game changer that gold tiles can be. Commerce snowballs as many techs boost commerce. Even if the final yield of towns is greater, they may never overcome the gold snowball. Early game gold boost opens up all kinds of wonder plays that are otherwise more dubious. Oracle -> deep sling being the obvious one, but also something like GLH I find is often prohibitively expensive because not only are you grabbing sailing early which may or may not be silly, but you have to grab masonry early and it's not even because you have a tile to quarry for a bonus resource. That no-cottage game where I raced to Monotheism for OR-fueled wonderspam is another example of gold-fueled shenanigans. Mid-game it can enable more greedy tech paths without jeopardizing TGL and liberalism. Most avg to below avg UUs are such because they don't fall along the normal bulb path, but high gold starts can make those UUs work out really well. Late game towns can overtake gold mines but only barely. I say only barely because town maximization locks you in to two civics. Also, late game food stops becoming restrictive (bio farms, corps) and you've probably stopped running slavery. Gold mines also get a boost from railroads.
I think not working capital gold mines is virtually always a mistake, because they get that snowball working so early, and the capital always has decent food, and bureaucracy gives a further bonus to gold mines. Other gold mines are situational depending on available food. In the example game settling that southern gold mine seems fine. However, I'd be very tempted to instead found a city that can work 3 gold and 1 silver mine and rush an academy there - pigs + FPs can support this food-wise. I never build academies. Ultimately, I like high commerce tiles because they let me have fun doing different things.