Why do we observe the nuclear force only in scatterings and decays? Why, at first glance, are the only forces we perceive to be gravity without quantizing, electromagnetism and nuclear forces only in disintegrations?
 A: 
Why, at first glance, are the only forces we perceive to be gravity without quantizing, electromagnetism and nuclear forces only in disintegrations?

If we accept as given the standard model of particle physics,SM with the addition of an effective  quantization gravity, it can be shown that it  mathematically describes/fits all of our observations and data.
Within this model your why is answered with: because we, as biological units, live in dimensions where h, the Planck constant is effectively zero, so quantization effects are not easily observable in our immediate surroundings. That is why it took so long to discover quantization.
In the SM, electromagnetism emerges  in our dimensions from the underlying quantum state, where the coupling constant is of infinite range. Gravity has a very weak coupling constant, and again of infinite range,  and emerges from the underlying effective quantized frame ( still to be proven). The infinite range in space is why we immediately perceive these two forces.
The weak and strong can be perceived in our dimensions with specific experiments exploring their short range.
This is the answer within the SM model for "why".

Why do we observe the nuclear force only in scatterings and decays? 

The nuclear force is a spill over force from the strong interaction, and is limited by the limited range of the interaction, as seen in the link given above:

Special experiments  allow to measure cleanly specific scatterings at the small   ranges that the strong force acts. The decays are not only due to the nuclear force, alpha decays,  but  also the weak interaction, beta decays,  which when quantum numbers and energy conservation allow can appear in decays of nuclei, and the same for the electromagnetic one, gamma decay.
The ultimate why is "why this standard model with this specific mathematics",  and the answer is : because it fits data and observations. 
A: Gravitational and electromagnetic potentials of a point source both fall of with the distance $r$ as $1/r$. This is a "long-range" characteristic compared to the effective nuclear forces that are mediated by massive particles (pions) and have a Yukawa potential, where the $1/r$ Coulomb behaviour is screened by an exponential $\mathrm{e}^{-mr}$. This makes nuclear forces effectively short-range.
As for "why" this is so, I discuss the origin of the Coulomb and Yukawa potentials from the underlying quantum field theories also in this answer, and in more mathematical detail in this answer for the electromagnetic form and here for forces mediated by massive particles.
