Gambesons ("padded armor") are extremely good at stopping broadhead arrows, though not so much for bodkin arrows. People often underestimate just how good armor really is. Probably because armor is treated like a cosmetic item in most fantasy media.
For example, wearing a gambeson with chainmail makes you essentially immune to slashing attacks. The real danger is broken bones or internal bleeding. The reason people still tend to prefer spears is because they're cheap, and the reason swords are so common is simply because, if you're gonna have a metal stick, might as well make it sharp. But ultimately, a sword is just a metal stick in a shape that's easy to use.
In an era where a single cut was often enough to get someone killed (from disease), having most armies suffer only 10~20% casualty rates from the vast majority of battles should be a pretty good indicator that armor is very good.
More specifically, it makes you effectively immune to the
slashing part of the slashing attack, converting it into blunt force and spreading the impact across a larger area, which diffuses the damage done significantly. Swords never really became truly
common, either, but as metallurgy advanced they did become used by more people since they became more affordable for the upper middle class (and also used less metal than an arming sword, because rapiers and smallswords, with their incredibly thin blades, became far more prevalent at this time), and by the Late Renaissance, even (wealthy) commoners could afford them. Spears remained by far the most common weapon, though, due to their unmatched ease of use/mass training, reach, andlow material cost.