The diatonic major scale is indeed 1-1-½-1-1-1-½. The intervals of a tone and a semitone also appear in other scales: an ascending melodic minor scale has 1-½-1-1-½-1-1: still five tone steps and two semitones. Your suggestion of 1-½-1-1-1-1-½ is used backwards in the descending melodic minor scale.
The arrangement of tones and semitones in the diatonic major scale is exactly the scale used in one of the chant modes: it's what is variously called the Lydian or Ionian mode (Mode 5) where the fourth, if it appears, is conventionally flattened. Mode 5 is used for the joyous chants of the church like the Salve Regina or the Alma Redemptoris Mater (links to YouTube), and they do indeed sound like major-key pieces. Chants in Mode 5 tend to have been composed later than others; the chants used in those videos are later than the chants contemporary with the words (11-12th century), which are in other modes.
The ascending melodic minor scale is akin to Mode 1 (usually called Dorian), where the sixth is often flattened if it appears.
As in many evolutions, it's difficult to tell when the major scale arrived, and hence form a view on its origins. However, as Mode 5 chants tend to have been composed later (there are more from the fourteenth and fifteenth centuries than say, the tenth), it's clear that it was emerging as a standard.
A very early example of a major-key work is Sumer is icumen in, written in England in the mid-13th century. The manuscript is written in chant-like notation:
The red cross indicates where the second voice in the round should start.
The answer to "What's the origin of the diatonic major scale?" is probably "It's impossible to say," other than that mediaeval musicians thought that particular scale and its arrangement of steps sounded good.
With regard to the arrangement of keys on the piano, they are arranged so that the major scale can be played on white keys only, starting on C: starting there produces the 1-1-½-1-1-1-½ sequence. There aren't any black keys "missing" — as Tim H commented, they were never there. The gap between what are called E and F is the semitone step in the scale. Black keys were inserted within the tone steps to divide those into semitones and create a chromatic scale.
Why that particular natural scale starts on a note called C and it wasn't called something else may be better addressed in another question, but I suspect that it has something to do with the white-note minor scale starting on what is called A.
Image: London, British Library, MS Harley 978, f. 11v, via Wikipedia