Pin bridges seem to have been taken over from harp designs, where the basic setup goes back to the middle ages. In that case the pins not only held the strings in, but provided a 'saddle' of sorts. They were often shaped in the manner of a sitar bridge, to produce a buzzing sound, and were called 'bray pins'. The harps were small, and probably tuned fairly low, so the buzzing helped the tone to cut through, rather like the way Flamenco players use fret buzz. Harp players use a special knot, but I've always found that a plain figure-8 knot works pretty well.
From what I've seen pin bridges were quite common except on Spanish guitars. The tie block bridge the Spaniards favored was something of a hold over from the lute bridge. On lutes the tie block pretty much is the bridge. I have seen one scholarly suggestion that the saddle on the guitar bridge originated as a sort of 'capo' on lutes; a way of shortening up the string length a bit. It may have become a standard part of the bridge when it proved to offer a more positive stop for the strings.
Contrary to the obvious inference, bridge pins are not wedges where the taper of the pin in meant to hold the string in place. Instead they are toggles. The string knot presses against the side of the pin and that holds it the pin in the hole. Making the pins a tight wedged fit in the holes is a good way to split the bridge. DAMHIKT
There are two common ways to implement a pin bridge. One is to use round pin holes and a slotted pin, and the other is to slot the bridge holes and use a round pin. The latter is far superior. A round pin in a slotted hole basically serves to hold the string in place until the knot seats on the inside. Once that has happened you can normally remove the pin and not lose the string. The taper of the pin just allows you to seat it snuggly so that it doesn't fall out. If the pin shoots out when you're tightening the string and kills the cat you didn't have the knot seated properly. I can't think of a gut/nylon stringed guitar that I've seen where the pin bridge was not slotted. That particular change seems to have emerged with low-end steel strings; its a lot easier to slot the pins than the bridge.
Steel string guitars, where pin bridges are normal, use a hardwood 'bridge plate' to keep the knots from cutting into the top.