@Adler - what is your location? Is the persimmon you saw growing in the wild? Planted landscape? Or is it nursery stock? Knowing your location ( general, not trying to violate privacy) helps when we offer advice. You can edit your profile to include location.
@GGB - east of the Mississippi River, there are only 2 trees native that are commonly called hornbeam. Carpinus caroliniana & Ostrya virginiana. The Carpinus is called variously blue beech, muscle wood, ironwood, and loose flowered hornbeam. Bark is smooth. Ostrya is called ironwood, and hop flowered hornbeam, it has a somewhat rough bark, definitely not smooth. And seed capsule is tight, much like a hop used in brewing. The two are easy to tell apart by bark and seed structure.
The leaf shapes of Carpinus, Ostrya, Fagus (beech), elm, & hackberry all are superficially similar, it takes a while to get familiar, but in reality all have different leaves. So I get the confusion.
Yes, I think North American persimmons should be used more than they have been in the past, they are native to all of USA east of Mississippi River and southern & Eastern Canada, so they are more hardy than the Asian species.