@DavidCorwin: Yes. In Browder's 1961 Annals paper "Torsion in H-spaces" his Theorem 6.11 says in particular that $\pi_2(X)=0$ if $X$ is a path-connected H-space whose homology groups $H_i(X)$ are finitely generated for each $i$ and nonzero for only finitely many $i$.

The proof in this paper seems far more complicated than the one in Hempel's book which is just a simple application of the loop theorem. This paper also uses the loop theorem and depends as well on something the author calls Partial Poincaré Duality. This doesn't seem to be a well known concept and the only reference given for it is the author's master's thesis. A mathscinet search turned up a published version in the 1975 Oxford Quarterly. It involves infinite cyclic coverings of manifolds and generalizes a paper of Milnor on infinite cyclic coverings. The statement is not simple.

Incidentally there is also Laudenbach's 1974 Astérisque volume "Topologie de la dimension trois: homotopie et isotopie" which proves this result for a broad class of 3-manifolds including $S^1\times S^2$, as stated on page 5. The companion question of whether homotopic diffeomorphisms are isotopic is also treated in depth.

You are right that there is no explicit statement here of the result you are looking for, so one has to read between the lines a bit. It does say that there are exactly two non-equivalent mappings $S^1\times S^2 \to S^2$ satisfying the appropriate homology condition, with one mapping obtained from the other by composing with the twist homeomorphism of $S^1\times S^2$, and this seems like the essential point. Surely he would have known what this meant for homotopy equivalences of $S^1\times S^2$.

@IgorBelegradek: The theorem of Sawashita appears to be incorrect in the case of $S^1\times S^2$. The kernel is ${\mathbb Z}_2$ rather than ${\mathbb Z}$ (and the extension is a product). The ${\mathbb Z}$ presumably comes from $\pi_1SO(2)$ but it should be $\pi_1SO(3)$, corresponding to a homeomorphism of $S^1\times S^2$ rotating $\{t\}\times S^2$ by the angle $t$. As in my comment to Andy Putman there is no difference between basepoint-preserving homotopy equivalences and those that do not preserve basepoint in this situation.

@AndyPutman: For $S^1\times S^2$ the mapping class group fixing a basepoint is the same as the one with the basepoint not fixed. This is because in the Birman exact sequence the map $\pi_1 {\rm Diff}(S^1\times S^2)\to\pi_1(S^1\times S^2)$ induced by evaluating diffeomorphisms at the basepoint is surjective. (For others: the Birman exact sequence is the exact sequence of homotopy groups for the fibration ${\rm Diff}(S^1\times S^2)\to S^1\times S^2$ given by evaluating at a basepoint, with fiber the basepoint-preserving diffeomorphisms.)

This paper uses the definition of the mapping class group as homotopy classes of homeomorphisms (or diffeomorphisms) rather than isotopy classes. Injectivity is fairly easy with this definition. More work is needed to get injectivity for isotopy classes.

If you want the incidence matrices to give the boundary maps in a chain complex computing the homology of $X$ then this is certainly possible and can be deduced as a special case of the general method for computing cellular chain complexes described in Section 2.2 of my algebraic topology book. In particular see the cellular boundary formula on page 140 which uses Proposition 2.20 on page 136. There is also an old book that develops homology theory from this point of view starting from scratch: "Homology of Cell Complexes" by G.E.Cooke and R.L.Finney, Princeton Univ. Press 1967.

This construction can be found in some textbooks on knot theory in the context of the unknotting number of a knot. For example it is described on pp. 58-59 of The Knot Book by Colin Adams and on p. 133 of Knot Theory by Charles Livingston. I have a dim memory of seeing it somewhere else as well. In Adams' book he gives an argument which is essentially the one in the answer by Wojowu below, except that Adams makes the unstated hypothesis that the starting point lies on the boundary of the convex hull of the knot projection, which simplifies his argument significantly.