If St. Matthew is principally concerned with describing Christ's generation according to the flesh (as has been said), why does he record our Lord's genealogy through Joseph and not through Mary?

First, it is important to realize that Christ's carnal descent (at least insofar as he is of David's line, thus fulfilling the prophecies¹) can be inferred even from Joseph's ancestry. This is because it was a custom among the Jews to marry into the same tribe and family.² That Mary, as well as Joseph, was of the house of David, can be seen from the fact that they both return to the city of David when ordered to return to their own cities for enrollment.³ Therefore, giving Joseph's genealogy is sufficient to show that Jesus was biologically of the House of David.

But why not just give Mary's genealogy? Here, St. Matthew demonstrates in a more subtle way how God "humbled himself to share in our humanity."⁴ For it was a custom among the Jews, in fact, a custom among most cultures, to trace genealogies through the father. The Holy Spirit inspired St. Matthew to follow this custom in order to show Christ's assumption of the whole human condition. Given that our Lord's carnal generation through Mary and her Davidic line could also be shown at the same time, by respecting the human custom of patriarchal genealogy the condescension of the Incarnation is all the more emphasized.
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1. See, for example, II Kings 7:12, 1 Chr. 17:11, Ps. 131:11, Jer. 23:5
2. c.f. Num 36:6-10
3. Luke 2:1-5
4. Roman Mass, Offertory Prayer