The problem is that there's no way to specify an exception specification like "I propagate everything that this lambda throws" (or, for generics, "that method M of class C throws").
interface Func<P, R, X extends Exception>
{
R func(P param) throws X;
}
In principle, one could imagine a syntax for variadic type parameters like
interface Func<P..., R, X... extends Exception>
{
R func(P... params) throws X...;
}