This question has another another approach to an answer which is taken in Section 6.3 of the book Nonabeian Algebraic Topology {NAT}. The basic idea is to form for a triple $(X,A,C)$ of spaces, i.e. $C \subseteq A \subseteq X$, and $C$ is thought of as a *set* of base points, a functor $\rho(X,A,C)$ which which in dimension $0$ is $C$; in dimension $1$ is $\pi_1(A,C)$, the fundamental groupoid of $A$ on a *set* of base points; and in dimension $2$ is the set of homotopy classes rel vertices of maps $$(I^2, \partial I^2, \partial \partial I^2) \to (X,A.C), $$where $\partial I^2$ is the boundary of the square $I^2$ and $\partial \partial I^2$ is the set of vertices of the square. You will notice that this structure does not form a group!

However it turns out that for each $c \in C$ the obvious map obvious map $\pi_2(X,A,c) \to \rho_2(X,A,C)$ is injective (Proposition 6.3.8). Further $\rho_2(X,A,C)$ has two compositions $+_1, +_2$ which make the whole structure into a *double groupoid* . There is also an extra structure of so-called *connections* which make it *equivalent* to the *crossed module* $$\partial: \pi_2(X,A,C)\to \pi_1(A,C).$$

Having got all this done Chapter 6 of NAT goes on to prove a 2-d Seifert Van Kampen Theorem for $\rho$ which by translation to the equivalent crossed module allows new calculations of 2nd relative homotopy groups, as pushouts of crossed modules.

The point is that the additional structures allow, once you have mastered them, clearer and more powerful proofs. For more discussion of the history and methodology, see this paper.