It is indeed fairly common knowledge, but I think we have to be pretty skeptical of the claim - it's a good example of how common knowledge can be based on just one appealing-sounding theory in an uncrowded field that then makes it into pop-culture.

There's a previous question on askhistorians that deals with this topic, but I don't think any of the replies properly answer the question.

A couple of years ago I was reading about the hysteria topic and learnt that the claim that vibrators were invented to cure female hysteria through orgasm was only based on one book, so I read this book, some of its sources and commentary on it, and came to the conclusion that there's no real evidence for it. So the answer to your question is that there aren't any accounts of doctors' reactions when they discovered what they were doing, because to my knowledge there aren't any accounts of doctors doing it full stop.

The theory originates from 'The Technology of Orgasm: "Hysteria," the Vibrator, and Women's Sexual Satisfaction' by Rachel Maines (1999), and all subsequent statements of the theory originate with this work. The grounding for Maines's theory that is well supported is that hysteria was indeed believed to be a real condition in the period she describes, and the vibrator was also invented in this period and used as a massager for various ailments, including hysteria. However the evidence Maines gives for the vibrator being invented (or used) to give women orgasms (or even 'hysterical paroxysms' that the doctors didn't know were orgasms) in order to treat hysteria is based on various statements from the time, by doctors, and manufacturers of vibrators, that could be interpreted to be implying a sexual use, but could also be interpreted as not.

For example the 2011 film Hysteria is an interpretation (ultimately inspired by Maines's book) of the life of Dr. Mortimer Granville, who Maines cites in ch. 4 as the "father of the modern electromechanical vibrator". Mortimer Granville wrote that he had never used his vibrator on women, and the vibrator was not intended for the treatment of hysteria, though Maines interprets his statement as being "defensive about the issue.", and implies that there was controversy at the time over their use in this way. Maines does have some more convincing quotes, the best one in my opinion is the following (from the beginning of chapter 4):

"In a discussion of electromedical technologies new in his day, in 1903 Samuel Howard Monell effectively summarized the demand of physicians since Hippocrates for some simple means of getting results with their hysterical patients: “Pelvic massage (in gynecology) has its brilliant advocates and they report wonderful results, but when practitioners must supply the skilled technic with their own fingers the method has no value to the majority.” For physicians in this line of work, the vibrator was a godsend: “Special applicators (motor driven) give practical value and office convenience to what otherwise is impractical."

This one indeed is quite suggestive, yet pelvic massage is a genuine therapy for pain in the pelvic area for men and women, so again, there is more than one interpretation. Maines's book has many other examples, yet none give any more concrete evidence for her theory than the Monell one in my opinion. Additionally, her assertion that masturbating women to orgasm was not seen as sexual seems unlikely given that, as she herself writes in chapter 3, the female orgasm and the role of the clitoris in causing it was known to the medical profession and discussed in many works even in Tudor and Stuart times.

Anyone interested in further reading should read Maines's book, of course, and also 'Galen and the widow. Towards a history of therapeutic masturbation in ancient gynaecology' (Helen King), which takes issue with Maines's claims about masturbation to orgasm as a treatment for female hysteria.