The TypesOf implementation you present works, in that TypesOf<[{value: number}, {value: string}]> does indeed evaluate to [number, string]. Mapped types on arrays an tuples result in arrays and tuples.
But there is an issue, reported at microsoft/TypeScript#27995 where the compiler does not realize that inside the mapped type implementation {[I in keyof T]: ...T[I]...} that I will only end up being the numeric-like indices. It thinks that maybe I will be things like "push" and "pop", and so T[I] cannot be assumed to be of type Argument:
export type TypesOf<T extends Argument[]> =
{ [I in keyof T]: T[I]["value"] };
// -----------------> ~~~~~~~~~~~~~
// Type '"value"' cannot be used to index type 'T[I]'.
Presumably this is why you asked this question in the first place. The GitHub issue is listed as a bug, but it's been open for a long time and it's not clear if anything will happen here.
In cases like these I tend to use the Extract<T, U> utility type to help convince the compiler that some type will be assignable to another. If you have the type T that you know will be assignable to U, but the compiler does not, you can use Extract<T, U> in place of T. When T is specified later, Extract<T, U> will evaluate to just T if you are right about the assignability. And Extract<T, U> will be seen by the compiler to be assignable to both T and U.
In our case, we know T[I] will be assignable to Argument but the compiler doesn't. So the following workaround using Extract<T[I], Argument> will suppress the compiler error without affecting the output of TypesOf:
export type TypesOf<T extends Argument[]> =
{ [I in keyof T]: Extract<T[I], Argument>["value"] }; // no error
type Z = TypesOf<[{ value: number }, { value: string }]>;
// type Z = [number, string]
Looks good!
Playground link to code