Archaeology is great and they do the best with what they have - which is often very little. Like other forensic sciences, archaeology attempts to piece together a story-line based on a limited series of clues. Often these clues can be interpreted in a variety of ways which can lead to heated debate on any given find. Other times, there are no clues available at all. What conclusions should be drawn in such cases? Should they be drawn at all? Despite the fact that it's logically understood that "lack of evidence is not evidence of lack," in the case of a (supposedly) major figure like David, there was a sense that we should have found something by now. When nothing was, it was taken as evidence that no such person actually existed.
What we see from this is, as is always the case, that whatever assumptions we bring to a given set of information will color our conclusions about it. It's interesting to note that the Federal Government continues to spend millions of dollars on the Search for Extra-Terrestrial Life (SETI) project despite its total failure to produce a scrap of evidence of alien intelligence. Why? Because it's simply assumed that given the vastness of space that there simply must exist other habitable regions that have produced intelligent life. No proof, but an accepted truth nonetheless.
Contrast this with the fact that the Jewish People have writings thousands of years old that describe the life and times of David in detail. These writings have been faithfully transmitted from teacher to student and from parent to child for a very long time. It's part of our national consciousness and has always been known to be true. How interesting that this counts for nothing in the eyes of the scientific and historical communities. Will they believe us more now that they see we knew this particular piece of information about our King long before they did? Probably not, since they already "know" that biblical accounts are largely mythological or historical fiction at best.
Perhaps in time Jewish national consciousness will be given its due and the opinions (and historical record) of the people who actually lived these events, knew its players and carefully transmitted it from generation to generation will be taken seriously. Until then, it appears that we'll need to patiently wait for science to catch up.
For a comprehensive look at the confluence of the biblical and archaeological records see Professor Ken Kitchen's On the Reliability of the Old Testament.