I think you're giving Dan Everett (the person you're describing) a short shrift here.
He says that his initial pro-recursion reports were attempting to fit Pirahã into the standard Chomskyian UG framework of the time. However, as he learned the language better, he started to realize that it didn't really fit and he was seeing the things he expected to see. For example, he initially described -sai as a nominializer (an "adapter" that lets other parts of speech be used in constructions that expect a noun). His later reports argue that it's actually not a nominalizer, but marks old information ("As you know Bob").
As far as I know, there is not actually a ton of non-Everett data on Pirahã.
The most well-known paper that disagrees with Everett is probably Nevins, Pesetsky and Rodrigues (2009), which reanalyzes Everett's published work. They start with a list of properties are are supposedly unique to Pirahã, and claim that many of these show up in familiar languages. However, I don't find that paper completely convincing. For example, Pirahã possessives can't be embedded: the equivalent of "John's brother's house" is ungrammatical and needs to be rendered as "John has a brother. The brother has a house." NPR argue that German has a similar restriction. This is true, sort of, in that "Hansens Auto" (Hans's car) is grammatical, but "*Hansens Autos Motor" ("Hans's car's motor") is not. However, this occurs because of a totally unrelated restriction: you can say something like "Hansens Vaters Frau" (Han's father's wife) because everyone involved is animate, and there's an alternate route involving "von" for embedding other kinds of possessives (as in "the motor of John's car"). Pirahã, as far as I know, does not have any possessive embedding at all, which does make it different from German.
I'm also not sure that cherry-picking individual examples from many other languages successfully refutes Pirahã's exceptionalism. German may prohibit embedding in this circumstance and Hindi in that one, but surely the interesting part is that Pirahã seems to prohibit it everywhere. As an analogy, I know people who do not write with their left hands, and other folks who don't write with their right hands. However, this doesn't imply that hands aren't involved in writing at all.
Realistically, this debate is going to be hard to settle without some more data. Unfortunately, Pirahã is said to be impossibly difficult to learn and there aren't a ton of native speakers wandering around universities. There is some data kicking around from another Australian language which may have similar properties, which might help.