You aren't shifting. The ratios of the harmonic series refer to intervals such as a perfect fifth (3:2) or a major third (5:4). They are not relative to the fundamental (5:1 or 3:1).
As to the keys on the piano, they use equal temperament, so the only truly "in tune" intervals are the octave. Since you can't adjust the tuning of the piano while playing like you can with most other instruments, this was developed to enable modulation to any other key where they would be equally slightly "out of tune". If you tune a keyboard using just intonation, which is the pure overtone series, certain keys couldn't be modulated to because they were unacceptably out of tune.
The reason that intervals get closer together the higher you get is because the harmonic seres is an arithmetic series (1×f, 2×f, 3×f, 4×f, 5×f, ...). In terms of frequency, the difference between consecutive harmonics is therefore constant and equal to the fundamental. But because human ears respond to sound nonlinearly, higher harmonics are perceived as "closer together" than lower ones.
Here is a handy graphic of the the overtone series which indicates that you are measuring intervals, not pitches.: