Why does stability only consider, whether the system is bounded? That doesn't mean the response is smooth? Why does (BIBO) stability only consider, whether the system is bounded? That doesn't mean the response is smooth?

That is, one could have system whose values are bounded. Still it could exhibit "chaotic" behaviour through its parameters.

So what is this kind of stability/instability concept really helpful for?
BIBO stability applies to linear systems with inputs and outputs.  See

https://en.wikipedia.org/wiki/Lyapunov_stability

There is a section on linear systems with inputs.  The article is clear and restating it here is unnecessary.