Independence from US & US allied weapons industry. They would have a lot less "red tape" to acquire ordnance.
Already owning and using Chinese weapons is a foot in the door towards acquiring more sensitive items.
I don't think there are major restrictions on the Gripen. For one a major part of the fleet is assembled by Embraer in Brazil. Another thing is that technology transfer was an additional factor that played into the decision of picking the Gripen over alternatives. On top of that comes the fact that Brazil ordered a sizeable fleet of around 48 jets. All JAS-39E/F standard, meaning the modern and useful version (lol).
In general, the Gripen is not only operated by fellow BRICS member South Africa, but also by Thailand. Thailand has had several exercises with China in the past were their JAS-39Cs met the J-10 and J-11.
All in all I don't think it's particularly vulnerable to geopolitical games and rather popular with countries that aren't directly part of the NATO sphere of influence. Put on top of that the fact that the US multiple times lobbied against it, like in Norway or Canada and you get the picture.
All of that makes it in my eyes very unlikely that Brazil would buy J-10CEs, which is why I think it's second grade rumor/gossip. I highly doubt tech transfer would take place or localized production, I also doubt China would in turn buy C-390s from Embraer, just like Sweden did.
As for the J-35, I think you missed my point. Which was that while it would bring something new to the table, it's much more expensive, politically charged and most importantly wholly unnecessary and thus doesn't justify the cost. There isn't a single country in Latin America that would warrant the pricey procurement and operation of stealth multirole fighters. Basically, they invested a lot in the Gripen, which outclasses everything in South America by a long shot and that's why neither the J-10 (redundant) nor the J-35 (unnecessary) are likely to be procured by Brazil.
On the note of Latin America though: am I the only one that hoped to see Argentina picking up the JF-17 or J-10 rather than ending up with hand-me-down F-16s as a consequence of their new goverment being rather obviously US pawns?