In the paper
R. Brown and P.J. Higgins ``Colimit theorems for relative homotopy groups'', J. Pure Appl. Algebra 22 (1981) 11-41.
we generalised to all dimensions the formulation and proof due to Crowell (1959) of the Seifert-van Kampen Theorem for the fundamental group. Our generalised proof did not involve singular homology, nor simplicial approximation. Consequences were:
The Brouwer Degree Theorem (determination for $ n \geqslant 1$ of $\pi_r(S^n)$ for $r \leqslant n$; and also the determination of $\pi_n(S^n \vee W)$ where $W$ is a wedge of circles (without using covering spaces);
the Relative Hurewicz Theorem (as a determination of $\pi_n(X \cup CA,x)$ if $(X,A)$ is $(n-1)$-connected);
J.H.C. Whitehead's determination of $\pi_2(X \cup e^2_\lambda,X,x)$ as the free crossed $\pi_1(X,x)$-module on the $2$-cells $e^2_\lambda$ (Whitehead's proof involved transversality and knot theory, and there are maybe three other proofs);
a generalisation of 3. to determine $\pi_2(X \cup_f CA, X,x)$ as a crossed module induced by $f_*: \pi_1(A,a) \to \pi_1(X,x)$ (for $A,X$ connected --at present no other proof is available). Note that Whitehead's result in 3. is the case $A$ is a wedge of circles.
The full story is also given in the EMS Tract Vol 15.
The proofs very much involve the use of filtered spaces, the algebra of the multiple compositions of cubes, and the relation with the classical invariants of relative homotopy groups.