I was wondering why we use the notes much later in the series then the first ones? Why do we rearrange them for a scale/ or put them in "order" to create a Major chord? Is it just how the keys are laid out so it will be more comfortable?
why do we rearrange them for a scale
Because a scale is all about playing notes in order of pitch. (Of course your diagram is also arranged in pitch order, but it doesn't contain all the notes in the full C major scale including all the repetitions within each octave.)
A scale is not about putting notes in order of importance - it's simply about laying them out so that from any given note, you can easily find where the other ones close to it are. In that sense, it is related to laying out the instrument so it's easy to play - it would probably be unintuitive to have a piano where the notes pitches jumped about rather than steadily increasing as you moved from left to right.
why do we ... put them in "order" to create a Major chord
I guess we typically say that a major chord is 'C E G', in that order, for similar reasons - because that's the order you see the notes on the keyboard.
You might notice, though that the major chord is (in a sense) a re-ordering of the notes according to the harmonic series, as it contains the notes C, E, and G - the first three that occur in your harmonic series.
How come we dont use the early overtones in order from a Harmonic series to create chords/scales?
There are lots of different scales and modes, and most of them do use some of the notes corresponding to the early overtones. But having notes corresponding to early overtones isn't the only criteria for a useful scale - we want to come up with a set of notes whose pitches create interesting harmonies and tensions with each other, not just with the overtone frequencies in the tonic (root) note.
Your table is essentially ignoring factors of 2. If you instead ignored factors of 3, the table would look quite different. However, our notename system is based on ignoring octaves.
As a note aside, organs don't just have 16" and 4" and 2" augmenting the regular 8" pipes registers, but also 5⅓" and 2⅔". Those registers are tuned in pure fifths with respect to the main notes, while the in-scale fifths in the main registers are tempered according to the temperament of tuning used for that organ.
Western music considers octaves a "noop" and keeps them pure. As the simplest interval, that is a reasonable choice. But it's interesting that an organ offers fifths as "pure" "noop" intervals tied to the keyboard keys as well.
So to get back to your question: CEG as a building block is stronger compacted and thus carrying semantics of its own than CCGCEGC which is not a condensed form of what appears in nature but a form reduced along the "octave equivalence" principle, reducing a natural phenomenon into a more symbolic form according to principles underlying our notation system.
Consider it like paintings which reduce 3D material to 2D material, thus making it easy to capture essentials with a much more compact form that reality, one that has expressive power of its own, particularly if you look at more remote forms like cubism or abstract art that appeal to our schooled aesthetics in spite of not being anywhere like "real life".
The simple answer is that Western Music is based on Fifths. The clarinet for example being based on the Chalemeau has overtones based around the fifth whilst the easier and louder saxophone is base on fourths, so the sax keys go up in octaves. To demonstrate how it works just go to the piano that starts on low A and play up in fifths you will find all the notes in the naturals followed by all the sharps. When you get to 13, oops! a clash so that is where the chromatic scale ends. Now collapse them all back to a single chromatic scale and you have what Confucius and Pythagoras worked out 2,500 years ago, they sound good together. Also note that the Bb in the C7 chord is not diatonic to the C scale but is a great leading tone because it is earlier in the harmonic series.