The question can actually be reduced to the following: "Why is the first harmonic the strongest?" I went into SE archives and found the following statements:
...and the amplitude of each harmonic in frequency space is proportional to its amplitude in the initial configuration in wavenumber space. Now the initial configuration for plucked instruments happens to be roughly what I depicted in the first figure in the Music.SE answer: something between a triangle and a sawtooth, and as we know both have a monotonically descreasing1 sequence of Fourier coefficients ($\mathcal{O}(\frac{1}{n^2})$ for triangle, $\mathcal{O}(\frac{1}{n})$ for sawtooth), so the fundamental does tend to have the strongest amplitude on the string in the beginning. [1]
leftaroundabout actually goes into some deeper explanation further linked in the post, so I advise you go there.
My small summary would be that with usual initial conditions, i.e. how you usually first strike the string, the first mode is has stronger than the others. For example, for a sine wave, the frequency is purely 1st mode, and as exemplified above, most other periodic shapes have their fundamental modes more prominent, although there can be some exceptions (also provided at [1])
(I also suspect that higher modes get damped faster too, but I couldn't find any sources on this, so it's only a speculation. Anybody that who can prove/refute this is more than welcome.)
Some relevant questions on SE:
[1] 1st answer at, "Does the Fundamental Frequency in a Vibrating String NOT Necessarily Have the Strongest Amplitude?", phys.se
[2] "Is it possible for a harmonic to be louder than the fundamental frequency?", phys.se
[3] "WHY do harmonics happen?", music.se
