An answer is given by the Relative Hurewicz Theorem, for which a well known form can be stated as follows:
If $(X,A)$ is an $(n-1)$-connected pair, then the pair $(X \cup CA,CA)$ is $(n-1)$-connected and the morphism induced by inclusion
$$\pi_n(X,A) \to \pi_n(X \cup CA,CA) \cong \pi_n(X \cup CA)$$
is given by factoring out the action of $\pi_1(A)$ on $\pi_n(X,A)$.
Note that this implies $X \cup CA$ is $(n-1)$-connected and so the absolute Hurewicz Theorem implies $\pi_n(X \cup CA) \cong H_n(X \cup CA)$; and if $(X,A)$ has the HEP, then the map $X \cup CA \to X/A$ is a homotopy equivalence.
A proof of this form is given in
R. Brown and P.J. Higgins, ``Colimit theorems for relative homotopy
groups'', J. Pure Appl. Algebra 22 (1981) 11-41,
and is shown to be a special case of a higher homotopy van Kampen Theorem, proved without using simplicial approximation or singular homology (but it uses a cubical higher homotopy groupoid!).