Henry Cotton might be who they're referring to. He was a psychiatrist who thought bacteria in teeth was the source of mental illness. Pulling teeth to cure the crazies became quite popular for a time. Not 100% that's what the magazine refers to though, since his "treatment" was not in any way exclusive to women. In fact, his surgical experiments on his own sons likely led to their suicides. Plural. As in, both his sons killed themselves.

I'm curious what the context for your quote is, can you post an excerpt of the magazine article, or link to it?

As far as the accuracy I've heard the factoid about dentures on QI and other places and can't find anything more definite for it. However, the attitude towards dentistry and dental hygiene has changed significantly over the last century. Orwell comments on teeth in The Road To Wigan Pier (1937):

The most obvious sign of under-nourishment is the badness of everybody’s teeth. In Lancashire you would have to look for a long time before you saw a working-class person with good natural teeth. Indeed, you see very few people with natural teeth at all, apart from the children; and even the children’s teeth have a frail bluish appearance which means, I suppose, calcium deficiency. Several dentists have told me that in industrial districts a person over thirty with any of his or her own teeth is coming to be an abnormality. In Wigan various people gave me their opinion that it is best to get shut of your teeth as early in life as possible. ‘Teeth is just a misery,’ one woman said to me.

So it seems reasonable that in that environment dentures would be a welcome present.

Here's a newspaper clipping from 1993 talking about an Amish dentist being sued for doing this, it had been going on with him for a decade. Outside of the Amish community, I'm unsure, but this is definitely well-accepted there; pulling teeth and using dentures is generally simpler than having to deal with more advance problems down the line.

Sorry it took me a bit to respond to this, but what you are likely hearing about is what's called a pre-nuptial extraction, or "Dowry Dentures" and yes it used to be a very common thing in depressed areas and in certain cultures. In fact, it still is practiced. I very quickly found a study from 2011 demonstrating that contemporary Acadian (French Canadian) dentists have performed them.

In Guatemala it was traditionally done because women were considered less able to withstand pain so families would spend the money to extract the teeth when a girl reached adulthood. Care was rare and disease was common.

I found a study of Dinka refugees describing the practice. Similarly to the reasons mentioned in Guatemala, this practice was thought to originate in a time when tetanus was epidemic.

Gold teeth are still popular in Central Asia and well-off rural families will have some of their children's perfectly good teeth replaced with gold ones.