How can I add a node on top of a rectangle path?

One possibility, although not very TikZ-idiomatic, is to combine your \path command with Torbjørn's suggestion into a macro:

\documentclass{article}

\usepackage{tikz}

\newcommand\namedrectangle[3][]{%
  \path[draw] (#2) rectangle (#3);
  \path (#3 |- #2) -- (#2) node[midway,above] {#1};
}

\begin{document}

\begin{tikzpicture}
\coordinate (upperleft) at (0,1);
\coordinate (lowerright) at (2,0);
\namedrectangle[Named rectangle]{upperleft}{lowerright}
\end{tikzpicture}

\end{document}

Another option. If you know corners coordinates use fit library to draw the rectangle while the text is added with label. You can use named coordinates or just numerical ones. In this last case include coordinates list inside brackets.

\documentclass[tikz,border=2mm]{standalone}

\usetikzlibrary{fit}

\begin{document}

\begin{tikzpicture}
\fill[red] (0,1) circle[radius=1pt] coordinate (upperleft);
\fill[red]  (2,0) circle[radius=1pt] coordinate (lowerright);
\node[draw, fit=(upperleft) (lowerright), inner sep=0pt, label={Named rectangle}] (rectangle) {};

\node[draw, fit={(3,0) (5,1.4)}, inner sep=0pt, label={[text width=2cm, align=center]Another named rectangle}
] (rectangle-b) {};

\end{tikzpicture}

\end{document}