I strongly prefer gender neutrality, and I think it is worth a little effort. I do understand that especially for non-native speakers, using he/him/his may seem very natural. However, we can fix the problem over time, if we choose to.
In this article, it would have been pretty easy to fix the remaining cases of inadvertent sexism, while remaining grammatically correct. I think some of these would be improvements even aside from the gender issues.
"the user willing to enhance his browser" -> "users willing to enhance their browsers"
"to develop his add-on" -> "to develop an add-on"
"the developer writes some code right in his browser, refreshes a web page to see the effect of his changes in the Jetpack extension code, then continues rewriting his code, and so on" -> "the developer writes some code right in a browser, refreshes a web page to see the effect of the changes in the Jetpack extension code, then continues rewriting the code, and so on"
"the user learns how to write his own GMail notifier" -> "the user learns how to write a GMail notifier"
"the user sees the whole Jetpack extension source code before he installs it" -> "the user sees the whole Jetpack extension source code before choosing to install it"
There may be documents where eliminating he/him/his would be awkward, but this doesn't seem to be one of them. Basically, I look at any inappropriate he/him/his as being a yellow flag that the sentence should be refactored.
By the way, I apologize for not having something to say about the contents of the article itself. I stopped using FireFox a while ago. I guess I'll just say that it seemed to cover the material pretty well.