Post History

$\newcommand{\Type}{\mathsf{Type}}$
$\newcommand{\El}{\mathsf{El}}$
$\newcommand{\U}{\mathsf{U}}$
$\newcommand{\T}{\mathsf{T}}$

You would not add a supertype to $\Type$, because $\Type$ is not part of the object theory. It is the meta-theoretic object that classifies (_all_) the types of your object theory.

If you want to be able to internalize certain families as first-class entities, this is often done via having universes _within_ the object theory. So, for a la Tarski universes, you add:
$$\U : \Type$$
$$\T : (\El(\U))\Type$$
Along with codes in $\El(\U)$ and a specification of how $\T$ decodes them. Then, you have:
$$\Pi x:A. \Pi y : B(x). \U : \Type$$
So you can have an $f$ with that type inside your theory, and it can act similarly to the families definable in LF. Essentially, $\U$ and $\T$ reflect some of the structure of $\Type$ and $\El$ into the object theory. Of course, $\U$ and $\T$ themselves cannot (consistently) be reflected within themselves, which often leads to a hierarchy of universes $\U_n$ and $\T_n$, each of which classify previous universes. All of these sit _within_ $\Type$ in LF, though.