Why did falcata style weapons disappear?

[Philippus Flavius Homovallumus]

that halberdiers will dead first if they face TRUE SPQR legionaries. They will become pilum cushions

Dismounted medieval knights didn't use any precursor missile weapons, as they think missile fight was "unchivalrous", so it was their obvious weakness if they face any skilled 2 handed weapons user, which in turns the 2 handed and without shields will easily pawned with crossbows...

Actually, dismounted knights often used Halbards themselves, this precluded the use of thrown weapons but in general knights were known to employvery varried fighting styles, making use of everything from twin shortswords to crossbows. As far as vulnerability to pila goes, early knights carried shields and later knights had armour which might well have been pilum proof. For the Romans facing halbadiers would be like facing armoured falxmen.