I could avoid some unsafe code if the types of functions and closures-without-captures implemented the Default trait.

Theoretically you could extend this to closures that only capture things implementing Default, but starting with only closures that have no captures seems like a safer bet, and it can always be extended later.

Example:

fn call_by_type<F: Fn() + Copy + Default>() {
    let f = F::default();
    f();
}

fn wrapper<F: Fn() + Copy + Default>(_f: F) {
    call_by_type::<F>();
}

fn foo() {
    println!("Hello, world!");
}

fn main() {
    wrapper(foo);
}