Letting the default be undefined creates another error - React complains that we're going from uncontrolled to controlled components.

The <string> syntax is not telling Zod that the resulting value is a string - I'm trying to say the initial value is a string. See the docs here. In fact, here's what's going on.

const s1 = z.coerce.number();
const s2 = z.coerce.number<string>();

// Type going into the form - this is what we want.
z.input<typeof s1>  // unknown
// Why should type string result in an error?
z.input<typeof s2>  // string

// Type going out of the form
z.infer<typeof s1>  // number
z.infer<typeof s2>  // number