Is the Bourbaki treatment of Set Theory outdated?

You may want to read this critical review by S.L. Segal of that essay:

https://zbmath.org/?q=an:00096766

As Segal points out, Mathias is mostly raging against the neglect of set theory, logic, and foundations as worthy subjects of study. In a further response to Segal's review Mathias admits that in his essay he was not attempting to be a 'sober historian'.

Altogether that essay is more or less a personal rant, not serious academic output. It is an invective against Bourbaki-influenced mathematicians for not taking logic seriously. It blames Bourbaki for the dismissive attitude towards mathematical logic and foundations that exists in the mathematical community. Mathias laments that Bourbaki did not deem Gödel's work as worthy of being included in a volume on set theory. This is what he means by Bourbaki's neglect of Gödel, not that Bourbaki's Set Theory is inconsistent, but that it's not an in-depth treatise on mathematical logic.

For academic purposes you can safely ignore any mathematical concerns in that essay and not be any worse off for it. You can likely find much more serious critiques of Bourbaki addressing similar concerns.

Bourbaki's treatment of set theory and foundational material is outdated. It's only meant to provide a solid starting point for the 'real math' in the subsequent volumes, not to study set theory in itself. For its own purpose it is entirely adequate.

One of the shortcomings of Bourbaki's Set Theory as foundations for math is that there is no mention of categories anywhere. Instead it uses rather contrived constructions such as 'structures' and 'species'. This language is now almost entirely extinct. Additionally, for ideas such as 'adjoint functors' there are no alternative constructions offered at all, and they are entirely absent throughout the volumes.

Don't read Bourbaki's Set Theory if you want to understand set theory as a mathematical field. Personal interest or wanting to understand Bourbaki's at-times arcane language are better reasons to look in that book. In any case it's not a very difficult read, but it contains extremely convoluted constructions for basic mathematical objects, and many of these constructions are meant to be forgotten once their existence is confirmed.

In fact for learning math it's probably a good idea not to read too much Bourbaki. Reading a few passages here and there is likely beneficial, but reading the entire series from beginning to end is probably a waste of time, as each of the general topics covered have advanced since and have their own modern canonical textbooks.

Mathias does not merely criticize a neglect of logic and set theory, but also documents serious errors in Bourbaki's Theory of sets. There are both conceptual errors, when an outdated formalism involving Hilbert's $\varepsilon$ is used throughout, and also technical errors when theorems are incorrect due to missing hypotheses. There is further confusion between language and meta-language. None of those involved in writing the volume and its sequels by Godement and others are professional logicians.

Instead Weil (and Bourbaki following him) relied on anecdotal mail exchanges with Rosset who apparently didn't pay too much serious attention to this. The $\varepsilon$ assumption turns out to be stronger than the usual form of the axiom of choice customarily assumed in ZFC. As Mathias puts it, it is somewhere between the usual AC and global choice. I find Mathias' conclusions concerning teaching logic in French highschools not that well supported, but his mathematical analysis of the mistakes in Theory of Sets and its sequels is spell-binding.

In this sense the Bourbaki treatment of set theory is not merely outdated; it is refuted.

Many of the participants in this discussion have read Mathias' "The ignorance of Bourbaki" but some may not have read his much longer piece "Hilbert, Bourbaki, and the scorning of logic". This can be found here.