The reason this is not workable is because in summer you also have to buy the fall trees and upgrade the axe up to steel, in order to get the fiddlehead fern. The final upgrade on the barn is 25K and the pig is 16K, and you need money for the fall seeds.
In the "need to spend to make money", I build the basic coop, get 4 chickens and make 4 mayo / day. This latest game, I got the coop in the last week of Spring, the barn in the 1st week of summer and the big barn in the 2nd week of summer.
It's definitely doable, I've done it several times myself. Animals are honestly not a great way to make money compared to crops. In my current playthrough, I planted maybe 20ish strawberry seeds in spring, and used the money from those plus some other crops and fishing to buy 64 blueberry plants and 64 melon seeds (plus another 64 melons later in summer, to replace them). Maybe 3/4 of the melons got put into preserve jars, I sold the blueberries immediately for quick cash. Then in fall I did 128 pumpkins and pickled almost all of them. This was plenty of money for everything I needed for the community center.
Okay, technically I didn't need a truffle on this run, because of the remixed bundles. But I got the deluxe coop built in early fall, and the vault done in late fall, and I ended winter with 90k gold. If I'd spent the coop upgrade money on barns instead (one day of fishing easily makes up the 4k difference in coop vs barn prices) and then spent the vault money on coop upgrades, and spent part of that 90k on the vault, I could definitely have gotten the barn in time and still done everything else.
(There's nothing particularly special about the numbers of plants, just trying to give you an idea of scale. I just did 128 spots because that's how much space I had where I could use quality sprinklers (which I got in late spring) on the beach farm, minus a few spots for scarecrows.)
P.S. if you get the bat cave instead of mushrooms, there's a chance you'll get a free pomegranate and you won't need to buy that tree. Don't count on it, though. I bought a pomegranate tree but not an apple tree in my last playthrough - again, though, that's maybe 1-2 more days of fishing, and there were definitely a few days I messed around doing random stuff.
For completing the Community Center!
More time than making skull cavern runs? I guess I figure you have to visit the secret woods at least a few times (potentially less now with mahogany seeds) and you could time those with traveling cart days. But I'm certainly not someone that does speed runs myself.
If you're talking about real-time based speedruns, those just go to year 2 and buy the red cabbage seed from Pierre's. If you just sleep a lot of days, it's a lot faster real time than either the cart or skull caverns. Plus, spending more in-game time lets speedrunners do a smaller number of sprinklers (i.e. less time mining) and still get enough cash for everything. Also they only make one trip to the secret woods, you only need 10 hardwood for the CC and that can be gotten on the same trip as the woodskip and fern.
The current CC speedrun route does check the cart a couple times for the pufferfish, but the chance of red cabbage being sold in just a couple checks is pretty low. Pufferfish has a really high chance of showing up in the cart for some reason.
I don't do speedruns either, for the record, I just watch TheHaboo on twitch a lot