What's the purpose of writing one's academic biography in the third person?
When I wrote a biography of myself, it was because someone in administration asked me to so they could put it into a prospectus, where there would be lots of biographies in a row all in the same editorial voice.
It was specifically required to be in the third person for that reason: all the profiles are presented as though they have been written by the omniscient author of the document. It wasn't written that way because I wanted it to be, but for an actual purpose.
Once written, that biography went on file and was put everywhere that needed one. That includes the university website, future grant applications, other advertising material, commendations for students, reports, papers, and sundry other places. Sometimes I copy it there, and sometimes it happens without my even being involved (sometimes without any actual person being involved!).
In particular, it's useful to put it up front on a public website so that collaborators can easily just copy accurate pre-written text in when they need to provide a profile of me for one of their own purposes.
There are places that I've written a first-person biography, which is generally a lot shorter. Those are places that are 1) in my voice, and 2) where it was worth the fuss of rewriting something tailored to the situation. In general, and particularly for things published by the university, my "official" biography just goes back in again, because it's there, it's been edited, and it's comprehensive enough.
I’d always considered it a means to achieve uniformity throughout the bios when one person is writing them. That is, if you have to write a short biography for each author of a paper, it makes more sense to do it in the third person.
Also, a lot of academic writing styles discourage the use of the first person (or used to). It’s more likely that the web page third person bios are a copy-paste, or simply written in a style to which the author is accustomed, rather than an attempt at a stealth brag.
They’re biographies, not autobiographies.
My This author's own opinion is that occasionally this is due to how ingrained 'academicspeak' is in academia. 3rd person / 1st person plural is the norm when writing journal papers.
In this sense, the reason may well be in fact the opposite to what you state: people avoid writing in the first person, in order not to come across as self-serving and arrogant and claiming personal recognition, but rather appear impartial and recounting events dispationately from a third-person perspective.
This, along with 'bio reuse' from contexts which specifically required the use of third person narrative, may be a reason why you might see such bios even in contexts where third person narrative would seem less relevant.