Yes, the writers are that stupid. Picard was born and raised in France but has an English accent. So why not his "secret son"?
After discovering that Adam Soong, Arik Soong, Noonien Soong, and Inigo Altan Soong are all visually and audibly identical to each other -- implying that, over the course of
400 fucking years, there is
absolutely zero genetic intermixing in the Soong family tree
what so ever -- Picard
also having a "secret son" is the very least of the in-your-face laziness on the part of this "creative" team.
I still, to this day, maintain that all the male "Soongs" are
just one guy. An immortal like Flint from TOS. Except that in Soong's case, he's just so unbelievably arrogant that, unlike other immortal characters, he doesn't bother with the hassle of changing identities every ~century or so.