Answer

I do not know whether the midrashim are trying to reconstruct history as it actually was. Midrash can also be written for educational purposes. And in general, even in Jewish law, derash is not bound by the rules of peshat, the plain meaning. The verses are written in a way that allows them to be interpreted both homiletically and according to the plain sense, so that we can learn from them the messages that emerge from both readings.
Although it is hard to accept that an aggadic midrash dealing with facts—what happened with Solomon—would simply ignore the facts entirely. Perhaps one can view derash as a kind of caricature. A caricature sometimes greatly enlarges one aspect (a mole, or some amusing distortion that a certain person has) and ignores all the other aspects. So it may be that Solomon, too, had such an aspect, but within the larger whole it was only one aspect. The plain meaning describes the whole picture, while the derash focuses on that aspect, which indeed existed, but was marginal or partial.
For example, “whoever says that David sinned is nothing but mistaken.” After all, I assume that the Sages were also clear that he sinned. They wanted to focus on the formal halakhic aspect: that he did not sin with a married woman, because anyone who went out to the wars of the house of David gave his wife a conditional bill of divorce. But the moral sin was certainly there. Rabbi Yaakov Medan discussed this in his book on the sin of David and Bathsheba. There you have a focus on one aspect that does not describe the whole picture, but is nevertheless correct in itself.