What steps did Brazil take to aid the Confederacy in OTL?
They granted belligerent rights to the Confederacy, allowing Confederate privateers to use their ports and sold supplies to them. In the aftermath of the Bahia Incident, they forced the Federals to return the Confederate vessel
Florida to the CSN (Although it was sunk in an "accident" before such could happen).
Economic ties do not guarantee alliances, strong economic ties didn't stop the Confederacy from trying to secede from the Union. The Confederacy had few economic ties with Mexico, and even fewer with France.
Indeed such does not promise ties, but we know for a fact such a desire did exist by the interest of the French to intervene in favor of the Confederates and their reliance on Southern cotton supplies. As far as economic ties go with Mexico, they were extremely important for the Northern provinces, which made quite a bit of money off trade with the South.
Plus you are ignoring my central point - Mexico and France would only ally with the Confederacy if it was to their advantage, and so far you have mentioned no advantage that Mexico or France would gain from this alliance.
I ignored no such thing, as I directly said it should be obvious; a surviving Confederacy serves as a useful buffer for the French efforts in Mexico, and the dual threat of a French-Confederate alliance will preclude American actions against Imperial Mexico.
An independent Confederacy would have less of a tariff wall than OTL's postbellum south.
The Confederates adopted the tariff rates of 1858 in 1861 and strengthened it in some ways with other fees.
The Confederacy did have its own domestic class of capitalists, but they had far less working capitol than was invested by northerners in OTL's postbelium south.
The value of the Plantation system was worth more than all the factories and railroads in the North, meaning they'd have plenty of capital to fund industrialization.
In OTL, roughly 1/7 the the Confederacy's black labor force and 1/10 of the Confederacy's white labor force served in the Union army, few who whom would voluntarily return to an independent Confederacy.
More than made up for by a reduction of casualties.
Lastly, you provide no explanation for how an independent Confederacy would suffer significantly less damage than in OTL.
If the war ends in 1862, for example, it should be quite clear the newly independent Confederacy would've avoided the damage inflicted over the course of the rest war. That there would be no damage to Richmond, destruction of Selma, or burning of Atlanta should be obvious, for just a few examples.
Your claims that an an independent Confederacy could have "50% of the U.S.'s production capabilities by 1900 or 1910" and "surpass the Union sometime in the 20th Century" are still contradicted by the
US Census of Manufacturing. It's like
postulating a world where Mexico had more industry than Germany or the Netherlands had more industry than France.
Comparing the census of manufacturing from IOTL, where the South was subjected to four years of warfare, had its capital class destroyed, a third of its working age men killed outright, lacked a protective tariff to stimulate its domestic industries, and finally had unfair shipment rates forced onto it, is simply a nonstarter. As I mentioned before, in 1860 the South had about 14% of the nation's industry and would only regain this position again by about 1910; presuming this rate of growth ATL, with no extensive damage to the South, would mean the Confederacy would have roughly a third of the U.S. total by 1910/1915. 30/70 = ~43% of ATL North/USA. Given the ATL Confederacy would definitely have a better growth rate, it's not at all implausible to imagine they'd reach 50% if not higher. Long term, they'd definitely do so, as OTL showed with regards to industrial development in the South since the 1960s; go look up all the car plants that have come to the South since the 70s or how the South is dominating in domestic textile manufacturoring nowadays.
I recounted a series of actual events, showing that while Britain and the US did become friends, it took an extended period of time and there were a lot of bumps along the road. How does a recounting of actual history make no sense? I did not mention the Great Rapprochement by name, but I clearly showed that Britain and the US did eventually become allies.
You answered your original question with this; it's more likely for the United States and Confederacy to peacefully resolve issues and become important economic partners than to become long term enemies.