I am left wondering how algorithmic the method described above actually is. Specifically, Kan states in his article A combinatorial definition of homotopy groups that the groups $N_k(X)$ need not be finitely generated. So it would seem this method is probably not effective/algorithmic in general. Is this assessment correct? If so, does the method somehow still give rise to an algorithm for computing the homotopy groups of spheres?

@Mark: Nevertheless, every compact manifold of dimension other than four admits a CW-structure (see mathoverflow.net/questions/36838). The question posed by Chris Gerig then makes sense precisely as stated. In fact, it is still interesting in dimension four, even if some 4-manifolds may not admit a CW-structure. Note: the question of CW-structures on 4-manifolds seems to be fairly open (see mathoverflow.net/questions/73428).

@Geoffroy: It was my pleasure.

@David: It does. Thank you very much.

The box product on $\textrm{Fun}(A,M)$ you mention is simply the Day convolution product coming from the symmetric monoidal structure on $A=(0\to 1)$ given by $a\otimes b=\min(a,b)=a\cdot b=a\wedge b$. Thus, under some conditions, the result you state that $\textrm{Fun}(A,M)$ inherits the pushout-product and monoid axioms from $M$ follows from propositions 2.2.15 and 2.2.16 in Sam Isaacson's thesis (linked in my answer). Do you know in what generality Mark Hovey proves this result?

(continuation) Most of the details in my answer are spent setting up all the hypotheses for the induction, and checking that the procedure above can be performed continuously when $\varphi$ varies over a (compact) space. In any case, I do hope that you will let me know if you find any issues or have any comments regarding my answer or this question. I will truly appreciate it!

@Tom: I hope the underlying argument is valid, even though I wrote it in a detailed manner not so conducive to intuition. The basic idea --- gleaned from Agol's answer --- is to take an isotopy $\varphi$ in the homotopy fibre, and use the isotopy extension lemma to inductively push the support of $\varphi$ outside of each stage of a suitable exhaustion of $M$ by compact submanifolds. This will converge to the constant identity isotopy because we are using the compact-open $C^1$ topology. (to be continued)