Pretty sure the weapons NS proved a piccie of are known as "bills", a weaponised gardening (specifically, hedge-trimming IIRC) tool that was something of an English specialty. Although remarkably similar hook-bladed polearms were used by the Italians and Ottomans, and the notoriously bewildering (and obscurely named) array of High and Late Medieval European polearms had no shortage of diverse designs which in various forms provided the same functionality. Give or take a few; staff-weapons could get awful specialised.

Main reasons for the general move towards powerful two-handed weaponry from around the 1300s onwards were on one hand their general versatility and effectiveness against cavalry, and the increasing ubiquitousness of good body armour among even common soldiery which on one hand rendered shields increasingly superfluous and on the other plain necessitated ramping up the raw destructive power employed to bust through.

Anyway, back to the falcata-kopis thingies. I was under the impression they were somewhat tricky to make and thus comparatively expensive for blades of their size? In any case, they didn't really fit into the Roman tactical system and as that pretty much dominated the entire Mediterranean military scene for centuries it is not surprising the concave blades fell by the wayside. Conversely the Romans' remaining troublesome neighbours - chiefly the diverse barbarians of Central Europe and the Iranians to the east - went for long straight two-edged blades instead, which the Romans eventually adopted too. Down the road you then had the assorted late Migration Period and Medieval single-edged choppers with a straight blade and flared edge, which AFAIK provided cleaving performance comparable to the falcata/kopis type while being structurally rather simpler and, hence, cheaper. (The first Medieval falchions may have been modified meat cleavers used by militiamen...)