Have you ever used a crossbow? I did for several years and still have one at home (in bad shape). There is plenty of truth it what you all say but you forget a lot of things. Of course, if you can get your hand on a full load of modern types, I'll take them but from what I read, I assumed that was not the case (actually, you'll find much more modrn bows around).
First, a crossbow is relatively heavy. Don't forget that fiberglass is not available any more. Then, you have to go with wood and steel for the mechanism. You end up with something which is heavier than a regular rifle and that you use more like a light machine gun.
Crossbow are also fairly tricky to make unless you go for very simple type with a barrel, a light bow and a small hold to maintain the string in place (then, you end-up with less kinetic power than with a bow, and much reduced range). Otherwise, you need to have a skilled man to make them. The quarrels are also more tricky to make than arrows. Bows are easier to make, and you can start with simple models. When your bow maker gets better he can make you long bows and finally the small highly maneuverable type used by mongols (almost a light machine gun).
I agree that you need less training but some people use bows from instinct. Just select the good guys and train them (arrows are easy to make). I also remind you that the british archers who decimated the french chivalry were lightly trained peasants.
Have you ever tried to bend a crossbow? First it takes time. Then, it takes a lot of strength, even for the smallest modern ones. If you have to make ones using woods, you'll end up with something that can't be bent without mechanical device. I'll leave you the crossbows and take the swords, axes, spears and morning stars (Banzaï).
With bows, you can train women and children to use them while the men wait for the shock.
Last and not the least important. bows can achieve indirect fire at a range of more than 200 yards. Therefore, I'll position my archers behind the walls (in second line) and fire arrows at you before you even can see me (not very accurate but I have plenty of supply and I'll clear your ranks). With crossbows you'll be using direct fire and your crew will be more exposed. In addition, with a bow I'll be able to fire 4-5 times before the ennemy get to my footmen (then they can continue to fire in direct support) while the crossbow will fire twice at most (In support of those same footmen your corssbowmen can do little because their weapon is too heavy and bulky and, then, your crews are killed trying to bend it once more
).
Actually, ideally, I would use both weapons. Bows as regular second line, used in fair concentration. Crossbows (exactly as described) as support weapons provided I can fire them from cover (the shield is a fun idea). I leave my footmen in first line waiting for the shock.
Why do you think crossbows never supplanted bows IRL?