Thanks to fmoreno and Gavin Taylor for kindly revising a first draft of this text.
TL;DR: The Amazon rainforest is greatly relevant for climate change. Some people have been blaming Brazilian exports in general for its deforestation. However, this is not true: when it comes to agricultural exports, the deforestation yields no relevant economic gains for the country, but responds for a significant portion of its carbon emissions. Though threatening the agricultural sector have shown some effectiveness in pressuring the government into taking action, this might end up misfiring. It'd be way easier just to ban exports from the Amazon region.
Preserving rainforests has been recognized as crucial to mitigating climate change, given their role ; the Coalition for Rainforest Nations (CfRN) has been pointed out as one of the most effective projects in this area by a Founder's Pledge report by J. Halstead and by SoGive analysis. When it comes to raiforests, the Amazon stands out, not only for because it’s the largest one, and one of the 9 tipping points of global climate regulation, but also because it has been a target of coordinated attacks: in 2019, São Paulo skies went dark due to huge forest fires caused by ranchers; in the last year, deforestation again reached new levels. Brazilian government has been at least considered negligent about this (if not openly supportive), displaying disposition to act only after international pressure and legal suits.
Thus J. Brice and T. Freitas Bloomberg’s article blaming Amazon deforestation on Brazilian livestock exports seemed at first quite plausible to me, and sounded a bell. They display a lot of data about the national economy, but almost no specific information about the regional economy. Similarly, French President Macron has recently stated that buying Brazilian soy implies cumplicity with Amazon deforestation. Ironically, we haven’t seen similar criticisms (at least recently) towards illegal timber and mining exports – which often cause the first stage of deforestation and are more likely to come from the Amazon region. I’ll give them this: the entire beef production should be traceable by now, the industry is morally abhorrent (besides animal suffering, JBS has been mired in legal scandals), and the government (and corporations) will only act if there’s economic and international pressure… but there’s an important flaw in their argument: productivity in that region is so low (because of the terrible climate, poor soil, lack of infrastructure an human capital…) that it responds for only a tiny portion of the country’s overall output. In both cases (beef and soy), most of the production still comes from the South and the Cerrado Region.
The truth is more complex and sad. Accurate models depicting how the boarders of the forest shift exist since the 90s, thanks to researchers Robert Schneider (then in World Bank) and Christopher Uhl, founder of Imazon – an important think tank that has been in the area to investigate the phenomenon for a long time, contributing to public policies that led deforestation to its lowest levels in the last decade. They call the regional economic cycles of deforestation “from boom to bust”: miners and loggers are usually the first to exploit forest areas (thus bringing basic infrastructure, such as simple roads), succeeded by land speculators and poor ranchers from other regions, attracted by low-price lands; they’ll put down the remaining forest to raise cattle and subsistence agriculture – with productivity decreasing each year, until they exhaust soil nutrients. In fact, these activities are extractive, since they depend on depleting scarce resources on the area. Consequently, the area gradually becomes more like a savannah (Cerrado) than a rainforest; the population will migrate to towns, so that the economy in the region is mostly dependent on government transfers and services – not agricultural exports.
A recently influential report linking deforestation in the Amazon to soy and beef production was published in Science (free draft version, with supporting material), claiming that "roughly 20% of soy exports and at least 17% of beef exports from Amazon and Cerrado" regions could be contaminated with illegal deforestation; because of it, Forbes and BBC then mistakenly announced that 1/5 of the entire national exports were so contaminated. However, if you really read the results, of the 13.6 million metric tons of soy exports from Brazil, only .5 (= 3.7%) were linked to Amazon illegal deforestation, and 1.4 million (= 10.3%) to Cerrado illegal deforestation. The Amazon biome responds for only 16% of all soy exported to Europe - way less than Cerrado (53%) and the Atlantic Rainforest (26.5%) areas.
When it comes to cattle, it’s even worse: the article remarks that, though Para exports are negligible, from 17.7 thousands of metric tons of beef exported from Mato Grosso and Para analyzed by the researchers, 4.6 million (=26%) may have been contaminated with illegal deforestation; but then, notice that (according to their fig. S30) Mato Grosso responds for only, 21.6% of Brazilian beef exports to European Countries (which totals 189.3k metric tons), and again, most of the production comes from the Cerrado region (figs S28 and S29): according to Table S15, only 7% of the national exports to Europe come from the entire Amazon Region. One could still argue about their methodology (given the difference of productivity between illegal deforestation areas and others), but I think this is enough to assert with certainty that this research gives no support to the claim that around 20% of Brazilian soy or beef exports (to Europe or anywhere) are linked to Amazon illegal deforestation – actually, this fraction is most certainly below 4% for both commodities.
Please, don’t misunderstand me: I love the Cerrado (and the Atlantic rainforest, and the Pampas, and the Pantanal…) as much as the Amazon forest. Nonetheless, the latter is way more important as a carbon (actually, due to longer seasonal droughts, Brazilian tropical seasonal forests might become net carbon emitters), global climate regulation and biodiversity. My point is that, by threatening to boycott Brazilian exports, it’s possible that other countries can successfully pressure the govern into taking effective action, up to a point; however, if the threat is not effective, blocking these products will likely have no direct effect on deforestation, and might possibly misfire – by impoverishing other farmers (and the whole country: agriculture was the only sector with positive economic growth in 2020) and causing bad reputation to preservation efforts - as warned by this ex-lobbyist. It has taken a while for Brazilian agricultural business (who have considerable political power and provided crucial support for the current government in the last elections) to realize that the fight for preserving the Amazon is not against them – that their interests actually conflict with those who are burning the forest, that the production from illegal deforestation is so low that there’s no reason to support it. I'm afraid some sort of general ban on Brazilian agricultural exports could jeopardize that. Maybe a moratorium concerning soy and beef from the Amazon region would be enough to settle this issue; even so, given that the first driver of deforestation is speculation with land prices (besides illegal timber and mining), I'm afraid such a ban wouldn't be enough to stop it.
P.S.: The most interesting material I’ve read about the history of deforestation in the region were published on Piaui Magazine; I strongly recommend the Arrabalde series, which sort of inspired to write this post – but it’s in Portuguese, and they often take a little while to translate their material into English (I could translate it, though, if so requested). When it comes to the latest news, Folha de Sao Paulo translated material might be useful. I tried to privilege material written in English (or translated) in this text, but this wasn't possible; if anyone has any doubts concerning one of these sources, I'll be available to answer them.
 Similarly, when asked about the importance of the Amazon region to Brazilian economy, Meg Smyngton, from the WWF’s Amazon Program, said “Agri-commodities are the biggest sector of the Brazilian economy in terms of exports. And agri-commodities are primarily behind what’s happening in the Amazon.” But notice the non sequitur: this does not at all implies that Brazil depends on agri-commodities from the Amazon region. Actually, the region output responds for only about 5.5% of Brazilian GDP.
 Take the Mato Grosso state as the worst-case example: it’s a region divided by the Amazon, Cerrado and Pantanal biomes, and it has the greatest agricultural output in the Amazon region. 80% of its illegal deforestation (which is likely to come from the Amazon region, since law places more restrictions there) come from only 2% of the farms. Imaflora estimates that a quarter of China soy imports from Brazil came from this state, and that 21% of these were likely to come from illegal deforestation – which means about 5% of all imports (including both Cerrado and Amazon).