A wright is someone who makes things.
A wright is a craftsman, someone who uses skills and experience to create something of use.
Examples: a shipwright, a millwright, a wheelwright.
It’s about craft and it’s often a suffix used for people who work with their hands.
And for playwrights.
Because playwrights, too, are using skills and experience to practice their craft.
I think, too, that creating music – being an arranger or composer – would be best described as a form of wright. Creating a musical work that is of use is a creative work, to be sure, but it is also a craft.
Think of it as songwright, a musicwright, a tunewright.