I was originally attracted to the movement as a way of finding ways of playing baroque music (the stuff that I spent most of my time with since I was a flutist) in a more intelligent way than I had been playing it. I have read the well-known treatises, and have applied, for various periods in my musical life, "rules" that I thought would help me to better understand and play in a more "authentic" manner. I learned to play baroque flute in order to play music written for my instrument in the way that it would have sounded when it was written. I feel very fortunate to have devoted ten years of my musical life almost exclusively to the baroque flute and the recorder.
Because I was learning at a time when there were very few experts in the baroque field, and those experts were around were still in the experimental stage of their lives, I learned most of what I knew (and I guess still know) from treatises. The main thing, in retrospect, that I learned from reading treatises (the "big" ones are by C.P.E. Bach, Rameau, Hotteterre, Quantz, and Leopold Mozart) is that good musicianship needs to always be guided by taste. Each of these writers had a pedagogical agenda and addressed specific problems connected with specific students (Quantz addressed specific problems that his student and patron Frederick the Great must have had), and each had his own unique ideas about musicianship.
Some people read these treatises as "rule books." I know now they are not. They are personal statements by accomplished musicians who were in positions prominent enough to get their books published. These are the books that the people against vibrato (PAV, maybe?) regard as authoritative evidence.
All I know about vibrato is what I know from practical experience. I rarely use vibrato when I play Medieval and Renaissance music with recorder players (as a string player). Recorder players play with very little vibrato, if any, because a diaphragm-centered vibrato, the kind that good singers and good wind players use, sounds very wide on the recorder. Of course there are recorder players like Michela Petri who have total control over the instrument, but for most mortals a vibrato that doesn't originate in the throat oscillates too slowly to make any kind of impact on the musical line (and a vibrato that originates in the throat sounds horrible), so when I'm playing violin or viola with recorder players I find that my sound blends better when I don't vibrate.
During the 17th and 18th centuries recorder players started using a fast finger vibrato called flattement to add expression. It is usually made with the index finger of the right hand moving in the air to the side of the recorder's finger hole. If I were playing a French trio sonata with a recorder player (or a baroque flute player) using flattement, I would certainly want to vibrate in order to match his or her sound.
Leopold Mozart warned against having a tremolo in the sound. I think that he was warning against the evils of a bad vibrato, both vocally and instrumentally. He probably meant to say that it is more tasteful to not vibrate at all than to vibrate tastelessly. It is impossible to teach a good vibrato in a treatise. A good string vibrato comes from a lot of care, practice, muscle control, and most of all a correct physical position on the instrument. String vibrato is very easy and very natural once a string player has a good physical relationship with the instrument. Getting that relationship to happen takes a lot of work, and the musical rewards are great. Why should we deprive ourslves of the ability to express ourselves with all we have for the sake of what some people call "authenticity?"