Somewhere in-between. In most ancient civilizations (Egypt, China, etc), as time goes forward we have more and more confidence that people were real. The earliest characters are allegorical/legendary, and later people were recorded in their own time. Ancient Israelite civilization is the same way.

It's interesting that you focused on David. From Biblical chronologies, he lived around the 11th century BCE. We don't have much of anything written from that period. However, we do get references to King David in the form "the House of David" (b[.y].t.d.w.d., which would be vocalized and pronounced in Modern Hebrew as bayit David) as the ruling dynasty in Judah (the Southern Kingdom; the name Israel refers to the Northern Kingdom and the United Kingdom that Saul, David, and Solomon ruled over) as early as the 9th century BCE. There's one possible reference to the House of David in the c. 840 BCE Mesha Stele and a pretty definite reference in the c. 800 BCE Tel Dan Stele.

This "house of (dynastic founder)" is a common way to refer to ruling families and at times even the states they control. For instance, the Ottomans were known in Turkish as the Osmanlı after their founder, Osman (Osman is the Turkish version of the Quranic name Uthman). Indeed, in the Mesha stele, there's a clear reference to Israel (the Northern Kingdom) being led by "the House of Omri", after one of the leaders of Israel mentioned in the Bible, King Omri.

It's possible that David could be a mythic founder unconnected with history and invented at some point in the intervening one hundred to hundred years before his supposed life and his appearance in the historical record (it's worth mentioning that neither of these are Israelite sources--Mesha is Moabite and Tel Dan is in Aramaic, probably left by the King of Damascus); however, I don't think it's particularly likely. I think the case that David was real is much more compelling than the vague possibility that he wasn't (there's nothing to suggest he was a mythic figure so doubters argue for the possibility of his being legendary, not that there's some clear mismatch between sources that suggests he was legendary). However, the degree to which the David recorded in the Hebrew Bible matches the historical David is impossible to judge, given the paucity of sources.

But yes, there's fairly good (all things considering) archeological and other evidence for David and many of the later Biblical leaders (particularly from the 9th and 8th centuries BCE onward). There's pretty meager evidence outside the Hebrew Bible for leaders before David. The best evidence we have is perhaps for a historical Moses and a historical Aaron based on much, much later priests who claimed descent from one of the two o them (see Friedman's Who Wrote the Bible) but it's nothing like the evidence we have for David. Earlier people and events (especially the conquest of Israel as told in the book of Joshua) in the Hebrew Bible tend not to fit into contemporary secular historical understandings. David may well be the earliest fairly well attested specific Israelite in the historical record. Wikipedia has a fairly good "List of artifacts in biblical archaeology", though keep in mind that most of these artifacts are valuable because they preserve some written record of the period.