When did it become customary to eschew gender pronouns in scientific essays? Anyone who's written a scientific essay knows that the use of he/she/him/her, etc are frowned-upon (except where necessary)

But where does this come from? When did it achieve (near) consensus? When were the guidelines first written by a major academic body?
While I don't know that there is one answer as to how this phenomenon came about (and I agree with you that it has gained traction in recent years), it has certainly been habit in some fields for a long time. Anthropology is an example of this, although I cannot find an American Anthropological Association style guide old enough that contains reference this (nowadays, AAA defaults to Chicago). My anthropology professors generally encouraged us to maintain gender neutrality, and the use of 'they' was considered an acceptable way to do so.

My graduate studies are in Women, Gender, and Sexuality Studies (WGSS) and, as you might assume, gender neutrality is generally encouraged. At the risk of stating the obvious, the use of 'him' or 'mankind' in generalized reference to people is considered inappropriately sexist. There are some circumstances where I have seen exceptions made to this-- especially in literature about sexual violence. Older monographs about sexual violence commonly refer to abusers and perpetrators by 'he' pronouns and victim/survivors by 'she' pronouns (see Why Does He Do That by Lundy Bancroft for an example). However, I have noticed that this is less and less true for books published since 2000-- probably the result of greater acknowledgement of male rape victim/survivors, as well as the occurrence of sexual violence among queer couples.

I also know that some WGSS writers explicitly refer to neutral third person references with 'she/her' pronouns, specifically in defiance of the academic tradition of using 'he/him' for neutral third person reference (e.g. "A doctor might tell his patients..." or "A police office is expected to holster his gun..."). A similar logic is used in defense of new words like 'herstory' instead of 'history,' especially when the material focus on/deals primarily with women. Indeed, I have seen pushback on the use of 'she/her' in this way on the premise that it, too, is sexist. The debate around this is very complicated and probably not going away any time soon.

I have seen folks (including on english.stackexchange) attribute the invention and/or rise in 'gender neutral' pronouns-- and even the invention of 'they'-- to the feminist movement, but even a brief foray into the history of the English language upends that argument (see my second reference). There may be evidence that gender neutral language is coming into greater usage and popularity in the 21st century thanks to feminists, but thus far I have only encountered the anecdotal. 

Outside of WGSS, I have seen writers switch between 'he' and 'she' for generic third person in a single piece of writing. For example, Bruce Perry does this in his book, The Boy Who Was Raised as a Dog, although some people may consider this to be a less formal piece of writing. 

None of the major style guides for humanities and social sciences (APA, Chicago, MLA) explicitly endorse 'they' as an acceptable third person singular pronoun (read more here). In fact, Chicago actively discourages it. While APA encourages 'gender neutrality,' apparently it considers 'they' to be 'too informal' for academic writing. 

In reference to your last question about guidelines, you may want to look through this page, which gives examples of gender neutral pronoun suggestions in chronological order, dating back to 1792! As others have argued (again, see my first reference), the use of 'they' is not some new-fangled trend, but actually has a long and complicated history. (An oft-cited, 400+ year-old example comes from Shakespeare.) So-called 'grammar nazis' who refuse the use of 'they' on the grounds that it is 'ungrammatical' may be surprised to find that the debate around the gender neutral third person in English is over a two hundred years old-- older, depending on who you ask. This doesn't stop some of them from being outright offended at the lack of subject-verb agree (or at least this is the most common objection I come across). Others argue that it strains their linguistic faculties to use 'they' as a gender-neutral third person singular (see linguist Sally McConnell-Ginet's chapter "'What's in a Name?' Social Labeling and Gender Practices" in The Handbook of Language and Gender [2003] for more on the 'awkwardness' of 'they' as third-person pronoun).

On a more personal level, I have a lot of feelings about the issue of 'they,' because I go by they/their pronouns. I have encountered a great deal of hostility in my day-to-day as a result of this. People have made all kinds of arguments to me about why it is wrong or why they can't cope with my pronouns. On a sociological level, it would certainly make for an interesting study.