How to get figure caption to span multiple pages, without having to switch everything to capt-of?

You don't need the capt-of package which is just one line and the same definition is in caption however caption boxes the caption text so if you want it to break you need to unbox it:

\documentclass{article}

% I can't have both caption and capt-of enabled.

% Caption alone doesn't let long captions span multiple pages.    
\usepackage[font={small,sf}, singlelinecheck=false]{caption}


\usepackage{lipsum}

\begin{document} 

\lipsum[1]

% I have many figures that are formatted like this.
\begin{figure}[h!]
   \centering
   \rule{0.3\textwidth}{0.3\textheight}
\caption[Short caption]{\lipsum[1]}
\end{figure}

\lipsum[1-10]

% This seems to be the best way to get long captions to span multiple pages, while being styled like the above figure.
\begin{center}

   \rule{0.5\textwidth}{0.75\textheight}

\bigskip
\setbox0\vbox{\makeatletter
\let\caption@rule\relax
\captionof{figure}[short caption]{\lipsum[1-4]}
\global\skip1\lastskip\unskip
\global\setbox1\lastbox
}
\unvbox0
\setbox0\hbox{\unhbox1\unskip\unskip\unpenalty
\global\setbox1\lastbox}
\unvbox1
\vskip\skip1



\end{center}

\end{document}