According to Protestantism, why God did not inspire New Testament writers to explicitly assert Sola Scriptura? I think it is a fairly uncontroversial claim that the Bible (and particularly the NT) does not explicitly assert Sola Scriptura (e.g. here, here).

Now, if God fully inspired the Bible (even to the point of inerrancy), it is fair to ask, from a Protestant point of view, why God did not inspire biblical authors to explicitly assert Sola Scriptura?

Although the answer "only God knows" is surely valid, theologically speaking, it is uninteresting. There must be several Protestant theologians (and perhaps early Reformers themselves) who have addressed this question, perhaps as an attempt to respond the obvious rebut "but Sola Scriptura is not in the Bible".
Protestantism, as understood by the reformers and their legacy, were fully comfortable with a systematic theology that proceeds from exegesis. They did not consider it necessary to be able to point to a specific text for every single doctrine if they observed that the Bible as a whole taught a particular doctrine. It actually makes the affirmations weaker if they are taught on a proof-text basis since it implies that the doctrine rises or falls on the interpretation of a single text, which hardly any biblical doctrine really does. It is relatively easy for heretics to create plausible (though invalid) arguments to poke holes in any of the common proof-texts that threaten their views. The systematic theologies of the Reformation tended to uses texts for evidence, but not in an exact text-to-doctrine fashion that misses the big picture in the details.

The doctrine of "sola scriptura" is a pithy formulation to fit into the "five sola" scheme, but to address it more technically they were really addressing the fact that God is only known via his divine accommodation of the creator to the creature through general and special revelation. They would then parse out some clarifications and make their case for the primacy of special revelation above general revelation.

The foundations for sola scriptura come down to the belief that God's revelation is the only way to reliably know about God and how he wishes to be pleased and what he desires from us, and therefore what constitutes a reliable source of divine revelation, which leads to scripture alone. 

The problem with questions of the form, "Why didn't God X" is that all of them are speculative. Unless God tells us his motivations, we don't know them and we would be fools to guess or to think that we can draw conclusions from them. There are boundless examples of things that we would think God would have done or not done if he really is this way or that, but those are not the ways that he did it.

Nevertheless, the reformers did not think they lacked a biblical basis for their sola scriptura beliefs. They didn't require explicit biblical statements to draw theological conclusions from the content of the Bible.