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The Bank allows you to reach a level of cash where you can throw away most gold-generating items in a run to focus solely on Heirlooms, which you need a ton of. So I'd say it's a good second District to get.
I do love the granary and will use it early in future playthroughs for sure, I didn't run the numbers like you did so nicely but I was also under the suspicion that the savings could be comparable. What I like about the extra food is the increased safety net in non-healer runs, which is very good.
First, get the (base) Hamlet built up to a point where it's fully functioning for maxed out Apprentice level heroes.
Second, get the 50 portraits together and build the Bank.
Lastly, do enough Antiquarian runs to accrue a banked sum of around 250K in gold. At that point, I imagine I don't need to do anything but grab Emeralds and above in gems and special item value, while devoting the rest of inventory space to heirlooms.
There's more math to be done, I'm not as familiar with the game and the practical outcomes as I could be.
For example, a critic of my argument could point out that, in a properly played game, you have a lot of spare money early on anyway, and so you're not really 'losing' that 20-40k your leaving in the Hamlet, you're just putting it to work for you, and that if you canabalize that money 10 or 15 weeks later, you'll still come out ahead.
The other districts do present a harder argument to make, because the Granary and the Bank are basically a long-term vs. short-term money comparison. It's almost a one-to-one comparison.
Puppet Theatre does save money on the Abbey and Tavern, but it's not that hard to use camping, the Jester or the Crusader to top off stress in the middle of a dungeon. As such, we end up talking as much about the opportunity benefit of not having to use (or even buy) those stress removal abilities, than we do about not visiting the Brothel. It's part gold-savings, and part combat enhancement.
The Cartographers Camp is similar, providing both combat and income benefits. Although the income benefits are questionable, when your inventory size is a much bigger limitation on gold gathering than your loot drop rate.
The Vinters is the hardest comparison to make, but it's also the one I find least interesting. In the start of the game, when you're doing lots of short missions, you'll get a modest stockpile of Blood. With proper play, you should be able to tread water decently enough, and be ready to tackle the Baron (or whomever) once you start hitting the critical mass of vampires where the Fanatic and running out of Blood start becoming unavoidable problems. It feels like a quality of life upgrade, but it's way too costly for that. But at least it's cheaper than the Bank, and having more blood to give to your Vampires who are Passive or Bloodlusting for buffs can be nice for boss fights.
This is my biggest problem with Bank-centric plans. Sure, you'll generate a ♥♥♥♥-ton of gold every week with 250k gold sitting in reserve...
Or you could just spend the 250k gold. Even if you send three people to the Sanitarium every week to both fix a locked negative quirk, and lock a positive quirk, it's going to take you a long time to spend that 250k gold, especially since your plan involves reducing Sanitarium costs to minimum.
The problem with the Bank is that the gold you 'invest' is just sitting there. You're not spending it. And as soon as you do spend it, your bank income goes down. The Granary provides you ~500 gold every week, without having to save a penny. There's just not enough things to spend money on that you really need that big upfront investment cost.
While I was using the Bank, I'd have to sell every trinket I didn't want as soon as I got it. I'd put off disease treatments or quirk fixes as long as possible in order to eek out every little bit of compound interest I could get. This made me weaker in the short term, because I wasn't just saying "Eh, I got gold to spare. Might as well shove these guys into the sanitarium to cure them right now." Granted, the early game is pretty easy, and it's a good time to make sacrifices. And when you reach the lategame, when things get hardest, that's when the Bank pays off.
But it does feel like there's not as much need to rush the Bank. It's a lot easier to get a stack of 100k gold in the end game than in the early game, and you could just buy the Bank as your second or third district just fine. You will still want to get it relatively early, since using the bank as your primary income source, instead of your inventory, does allow you to stockpile heirlooms for the rest of the districts. I just think that since it's returns are based on your initial capital, you might as well earn that capital later on, when tens of thousands of gold is easier to come across.
Yeah my math skills have seriously deteriorated over the years. It's always helpful and pretty impressive to see the time and consideration put into things like this, though. I remember playing Wasteland 2 and coming across these fantastic charts by this one player that clearly demonstrated why you should always use weapon X or stat array Y. It was an extremely convincing argument, :)