Is sinusoids the nature's default signals?
Time invariance plays a huge role in nature. Most systems (including your ear/brain) don't have an absolute time reference but treat all points in time equally. That results in a preference for the description of these systems with essentially time invariant basis functions, which is what (complex) sinusoids are.

For linear time invariant systems, the complex exponentials are even eigenfunctions of the system and allow you to describe the system as a "diagonal" operator, i.e. in a very simple way. That even includes a system that just performs a time delay.

Nonlinear time invariant systems can also be described in terms of sinusoids, but it gets way more complicated there. A nonlinear time invariant system excited by a periodic input will produce a periodic output in general, and you can analyze both in terms of sinusoids. 

But you can also perform a more mathematical rigorous analysis by looking at the nonlinear system response to small disturbances of any input signal. And taking sinusoids to describe these disturbances gives you a local description in a time invariant way.

So all in all, if you have a system that is time invariant, no matter of what nature, time shift invariant basis functions are helpful, and complex sinusoids are precisely that.