Cloth and hide armour manufactured on layered absorbing principle is probably the most widespread style of pre-modern armour though. I do know that kevlar isn't satisfactory against blades, but it certainly seems that armours of different materials did work when using roughly same principle of manufacture. That being said, I have never seen explict historical refrence to silken armour (other than as decoration or such), at least I don't remember any off the top of my head, though there are some anecdotal stories about Mongols relating to that.
Plant fiber, hide and even paper are all well attested though.
Yes, those were incredibly proliferate. Quilted cloth, gambeson, and in most cases where even metal armor is available, just layers of cloth would be used underneath it was commonly used whenever people could get their hands on it. But while it can offer resistance to a person with a pointy stick, sword, or axe, it is highly unlikely that it would offer very much protection against pointed projectiles without any other sort of stiff protection. A person would have to wear a lot of it for it to offer the protection a piece of treated plate the medievals used, to the point where it inhibits the person's movements.
Silk can be quite strong, but when you layer it to that amount, the material you're using isn't going to be all that significant.
The strength of kevlar and silk isn't in its ability to resist shears, it's from the force needed to snap it when you try to pull a strand of it apart. It's tensile strength is incredibly high like spider silk for example. While this is very good for absorbing blunt kinetic energy like deforming bullets, it isn't very good when you take a blade to it and apply shearing force. Against something with a pointed tip, the bolt is going to punch a hole through it forcing the fibres apart as it goes further and further. Against a hunting broad head, it's not going to do very much good unless you have an unwieldy amount a lot of it.
Going off topic, even if silk was abundant, I don't think it would be used as much as plant fibre or mammalian fibre cloth simply because it's not meant for any heavy duty sort of work, or at least when not chemically treated. Silk has a tendency to lose a significant amount of strength when wet for instance, when the person is sweating and doing vigorous exercise it's likely the silk isn't going to hold up, likewise, in the rain or in a moist environment, silk likely wouldn't hold up very well rough treatment in general.