Why isn't "holek" instead of "devcat" a proper solution?
Though there can be some difference between words děvče and holka. Děvče is always a girl. Holka can by also a girlfriend, it can be an expressiway way to address adult women (but děvčata as well) and it can mean a prostitute.
One would think this sentence could be translated without "těch." Does Czech now have a definite determiner?
No, without těch it would mean you need the names of (some) 19 girls, not of the specific 19 ones.