No, you are correct, archers are stronger than crossbowmen against light melee infantry, such as spearmen and landsknecht.
As they do a total of 14 Damage (2 more damage than crossbows)
With a higher Rate of Fire.
Thus their DPS against Light melee Infantry is 9.3 Damage per Second.
vs a Crossbowman against Light melee Infantry is 6 Damage per Second
In a Archer vs Crossbow in the current game mechanics where none of them have bonus vs each other
Archer has 4.6 DPS against a Crossbowman who has 6 DPS against an archer.
So in an archer vs crossbow, the crossbow wins.
Exception is Arbaletier that has +5 range armor with deploying Pavises (Making them exceptionally strong vs archers)
Or longbow men that outrange Crossbowmen.
The reason why you rather stick with Crossbowmen with how the game currently is, is simply because Crossbows are just far more Universal. While they don’t do as much DPS against spearmen as a archer would do, they do enough damage regardless, but more importantly, provide that extra DPS support against heavy armor units such as MaA and Knights, something archers won’t be able to do, and currently crossbows are stronger than archers in a range fight against each other.
Now for my suggestion on adding crossbow to feudal age, also means a buff to archers.
Meaning archers +7 damage against Light Melee Infantry, should be +7 damage against Light infantry in general.
Meaning that Archers should be doing 14 damage against a crossbow who does 12. so in a crossbow vs archer fight, the archers should always win (Exception being french unique crossbow the Arbaletier)
This against should lean back to the core balance triangle i mentioned earlier:
Spears > Horsmen, Knights
Archers > Crossbowmen, Spears
Horsemen > Archers, Crossbowmen
Man at Arms > Spears, Archers
Knights > Man at Arms, Archers.
Crossbowmen > Knights, Man at Arms.
This is just a list of Hard-counters. I.E units that counter specific units really well for related to their cost.
While there are Neutral points such as Horseman vs Knights, the Knights would obviously win against a Horseman. Due to damage and armor difference. However it is not a hardcounter unit against Horsemen.
As the time it takes for a knight to kill a horseman is almost tripple of what it takes for a spearman to kill a horseman.
By giving archers +7 bonus damage against Light-infantry, they also become a viable cost effective option against Handcannoneer spam, and would help greatly at keeping them at bay. Handcannoneers are among the strongest unit in the game, but is quite costly and a rather slow moving unit, making up for their strenght.