You'll find a different looking approach in my paper: C. D. Godsil, Matchings and walks in graphs, J. Graph Theory, 5, (1981) 285–297. The argument there shows that if $G$ is a graph with maximum valency $k$, then there is a tree $T$ with maximum valency $k$ such that that the matching polynomial of $G$ gives that of $T$. Sice for trees, the matching and characteristic polynomials coincide, the bound on zeros of the matching polynomial follows from standard bounds on the spectral radius of a tree.

I would not say that the argument in Heilman and Lieb has suffered due to the passage of time.

You'll find a different looking approach in my paper: C. D. Godsil, Matchings and walks in graphs, J. Graph Theory, 5, (1981) 285–297. The argument there shows that if $G$ is a graph with maximum valency $k$, then there is a tree $T$ with maximum valency $k$ such that that the matching polynomial of $G$ divides that of $T$. Since for trees, the matching and characteristic polynomials coincide, the bound on zeros of the matching polynomial follows from standard bounds on the spectral radius of a tree.

I would not say that the argument in Heilman and Lieb has suffered due to the passage of time.