Latex Beamer with overlays and framebreaks inside a theorem environement

Question

I am currently working on a Latex Beamer presentation and would like to uncover my proofs sentence by sentence to ensure a digestible pace of presentation. Some of my proofs are longer than one slide. I know that beamer does not support overlays in combination with any kind of framebreak, manual or automatic but I can't be the only one to face this issue, right?

Of course, hacky stuff like beginning a new proof environment and hiding the unnecessary QEDs and Headlines or redefining some of the underlying code of the beamer class come to mind but if there is an easier option, that would be very welcome.

What I would like to have working is something like this:

\documentclass[aspectratio=169,10pt]{beamer}
\usepackage[utf8]{inputenc}
\usepackage{amsthm}
\usepackage[ngerman]{babel}
\newcommand*{\theorembreak}{\usebeamertemplate{theorem end}\framebreak\usebeamertemplate{theorem begin}}
\newtheorem{satz}{Satz}
\setbeamertemplate{theorems}[numbered]
\begin{document}
    \section{Vergleich mit Verzweigungsprozessen}
    \begin{frame}[allowframbreaks]
        \frametitle{\secname}
        \begin{satz}[Stochastische Dominanz der Clustergröße]
            Some Text
            \begin{equation}
                some equation \tag{4.2.1}
            \end{equation}
        \end{satz}
        \begin{proof}
            \uncover<2->{More Text.} \uncover<3->{Even More Text.} \theorembreak
            \uncover<4->{So much text, it doesn't fit on the first page.}
        \end{proof}
    \end{frame}
\end{document