By the time you have a perfectly reasonable model of a system that is good enough such that computing the transfer function’s poles actually tell you something interesting about the system, there’s way more you can say about the system than “it is stable.”
There are maybe some lessons to be drawn from basic “classical” control theory, but many are better stated by just analyzing the system directly.
(As a side note, I’m not saying there’s Zero value in analyzing transfer functions, just that it’s a long way to the top from there.)
By the time you have a perfectly reasonable model of a system that is good enough such that computing the transfer function’s poles actually tell you something interesting about the system, there’s way more you can say about the system than “it is stable.”
There are maybe some lessons to be drawn from basic “classical” control theory, but many are better stated by just analyzing the system directly.
(As a side note, I’m not saying there’s Zero value in analyzing transfer functions, just that it’s a long way to the top from there.)