There are a few different elements of this question that I think might be best addressed a bit separately that I think might make the whole make more sense.
"When the UK and France partitioned The Ottoman Levant"
There was no singular partitioning of the Ottoman Levant, and the partitioning did not happen just between the UK and France. The most famous partitioning between the UK and France, the Sykes-Picot Agreement, was not actually how the region was partitioned. This is perhaps most notable in northern Iraq, which Sykes-Picot assigns to the French, but also in the "zones" concept invented the idea that some areas would be under direct French or British control while the hinterlands would be semi-autonomous spheres of influence.
For the purposes of your question, however, Sykes-Picot actually did end up being a fairly close representation of how the boundary between what would eventually be French territory and what would eventually be English territory were drawn up.
Why?
Well, Mark Sykes' negotiating position, as he announced to the British cabinet, was to draw a line on a map from the "E" in "Acre" (today in northern Israel) to the last "K" in Kirkuk (today, as fate would have it, a city that is essentially on the "border" between Federal Iraq and Iraqi Kurdistan.)
I have never come across any real suggestion as to why Sykes picked these two entirely arbitrary cities from which to draw this line. And the Sykes-Picot agreement line doesn't follow it exactly. But the line that was selected until you get to Iraq is essentially the divide between Lebanon/Syria on the northern side and Palestine/Jordan to the South.
So in the case of the Levant, (and without getting into the longer story of why the Ottoman province of Mosul ended up as part of British Mandatory Iraq), the dividing line was basically an arbitrary negotiating position.
If the French intended to take the areas with Mediterranean maritime access
I haven't actually come across any suggestion that this was part of French demands or thinking at all. French claims in the Levant were based on a few things:
By contrast, British strategic goals had a lot more to do Mediterranean maritime access: they explicitly wanted control of a buffer territory in Palestine that could protect the Suez Canal.
why did the UK get Palestine/Transjordan?
Of these, Palestine is the most interesting one. The actual final agreement was not only between the French and the British (Sykes-Picot), but also included the Russians (Sykes-Picot-Sazonov). Between the three of them, they couldn't agree who had the right to control the Holy Land, so they just shaded it in brown and said that it would be an "international zone." The reality at the end of the war, however, was the the Russians were out of the war entirely, and the British were in by far the stronger position to make demands because they, along with their Arab allies, had done virtually all of the fighting in the Palestine campaign, to which the French had contributed almost nothing.
As a result, the Mandate of Palestine was awarded by the League of Nations to Britain.
For a much longer explanation of many parts of this, I actually did a couple /r/askhistorians podcast episodes a couple years ago that might be of interest: