Putting equations in \intertext to annotate a proof

This appears to just work, depending on what you want he output to look like, did you get an error message from your use of \intertext?

\documentclass{article}
\usepackage{amsmath}

\begin{document}

\begin{align}
1 &=2\\
\intertext{aaa}
a+b&=c+d\\
\intertext{dont't forget that
% use primitive display math here as a nested equation
% disturbs align counting
$$1111=1
\refstepcounter{equation}\eqno(\theequation)$$}
0.5&=\frac{1}{2}
\end{align}

\end{document}

The code above produces the following output:

You can put $<equation>$ in \intertext, as demonstrated below

\documentclass{article}
\usepackage{amsmath}

\begin{document}
\begin{align*}
    f(x) & = x^2+5x+9\\
         &= \left( x+\frac{5}{2} \right)^2-\frac{25}{4}+9
\intertext{because $\left( \frac{5}{2} \right)^2=\frac{25}{4}$}
    f(x)    & = \left( x+\frac{5}{2} \right)^2+\frac{11}{4}
\end{align*}
\end{document}

In answer to the question

Otherwise, what do people generally do when they have an equation that's important to the proof but not part of the direct logic?

I suggest using the extension witharrows.

\documentclass{article}
\usepackage{witharrows}
\begin{document}
\begin{DispWithArrows*}
    f(x) & = x^2+5x+9
           \Arrow{because $\left( \frac{5}{2} \right)^2=\frac{25}{4}$} \\
         & = \left( x+\frac{5}{2} \right)^2-\frac{25}{4}+9 \\
         & = \left( x+\frac{5}{2} \right)^2+\frac{11}{4}
\end{DispWithArrows*}
\end{document}