I favor 'Ain Qedeis, 80 kil. south of Beer-Sheba, as the Qadesh of
the Patriarchal narratives. It preserves the right name and agrees
with a literary/topographic analysis. It is nothing more than an
accident of survival that no source earlier than the Hebrew Bible
mentions 'Ain Qedeis on the edge of southern Israel, for it most
certainly did exist in patriarchal times along with two other springs
in the area: 'Ain el-Qureirat and 'Ain Qoseimeh. We are in no need of
secular historical evidence here, although epigraphy would be nice,
since the location of 'Ain Qedeis matches the biblical geography and
Qadesh is reflected in the modern name of the site. To go looking for
the Qadesh of Genesis 20:1 in Lebanon makes no sense, at all.
Similarly, the modern Darb el-Shur extending from Hebron to Egypt
must be, in the economy of assumptions, the first road to travel in
any search of the biblical Shur that was situated south of Qadesh. To
say we should look north for a place located south can't be taken
seriously given the alternatives.
It is certainly no contradiction of the sacred text to say Hagar's
wilderness was called Beer-Sheba in retrospect by the biblical
author. Everyone knows, or should know by now, Christopher did not
discover a place called "America". You say it defies credulity that
Abraham would eventually end up where Hagar found water. Why? If
there was no water there until Hagar came there definitely would be
people, yes conceivably proto-Philistines from the coast and why not
Abraham ahead of these, coming there immediately after it became
known. Your attempt to make the opening words of Genesis 21:22, and
what follows it, perfectly synchronous with Hagar's good fortune
seems to me like a bit of special pleading, since events happening
back-to-back may also be described with the statement "it came to
pass at that time."
You ask: "Wouldn't a Hebrew author portray the proto-Philistines as
being foreigners to Canaan?" Maybe if he was trying to support your
theory. The name Abi-Melek says absolutely nothing about the origins
of Abi-Melek, or even his native language. Same with Phicol. This is
outdated 19th century scholarship.
Abraham moved from Bethel to Mamre, in Hebron, according to Gen.
13:18. Here was still in Mamre the day before the destruction of
Sodom, and there is nothing in Gen. chs. 18-22 about him supposedly
going back to Bethel to witness this destruction. In Gen. 18:33 he is
returning to Mamre. In Gen. 19:27 he wakes up and returns to the spot
in Gen. 18:22, on the line between Mamre and Sodom, to see the smoke
of Sodom. In Gen. 20:1 he is leaving Mamre and heading SOUTH. He
makes a stop somewhere between Qadesh ('Ain Qedeis) and Shur, and
eventually moves on to Gerar.