That's a great example of the rippling social effects of technology. The reason swords could shift from cutting to stabbing was improvements in metallurgy.
To make a rapier that will hold a point and won't either bend or break on the first solid thrust, you need steel-making tech that wasn't widely available in Europe until the Renaissance.
Most medieval swords were really more like edged clubs; the
narrowness of the edge let you deliver a lot of force to a small area,
but it wasn't "sharp" as we'd understand the term.