Take a super-ultra-maximalist version of Solomon's reign and draw it on a map. The "Empire" we get is small compared to known ME empires, and half of that size

Take a super-ultra-maximalist version of Solomon's reign and draw it on a map. The "Empire" we get is small compared to known ME empires, and half of that size is because such a version could go up to Tadmor (Palmyra). If we apply a more realistic (IMHO) version, we get something tiny by Imperial standards. In the same token, even a straight reading of the text should never give us the term "Empire" regarding David. His Kingdom fights against the very close neighbours and often wins, but is prone to drought and civil war. Even winning fights against close neighbours isn't much of an indication of Empire - when the US wants something from Mexico, it doesn't have to fight for it. So I agree with you, but I think half of the exaggeration is due to more modern readings and (in this case) strawmen invented by people looking for something to knock down - the only person referring to "David's Empire" in the linked text is... Finklestein.