>You either identify the transfer function model using step/impulse tests
This is what I tried (step, not impulse). Did not work well. The result was too dependent on the "starting position".
The course professor probably knew less about controls than I did. He was a computer engineering professor.
> The result was too dependent on the "starting position".
PID controllers always control around a nominal operating point (which you pick), which is why deviation variables are typically used in the mathematical derivation. That's just the basis of the control algorithm.
2. Yes, but the transfer function is usually only to simulate the system in Simulink or some such software. In the real-world, you never have the transfer function.
So how do people do it?
You either identify the transfer function model using step/impulse tests (covered in undergrad courses and not that hard to do -- the least sophisticated way is to just measure the response to a step and graphically estimate parameters K and tau; it gives you a good starting point), or you can tune a controller by hand starting with a tuning heuristic and then refining. https://en.wikipedia.org/wiki/PID_controller#Loop_tuning
More advanced ways of identifying the transfer function is to collect data (either using pseudorandom binary sequence, i.e. PRBS step tests, or just simple plant step tests) and then doing "system identification", that is, fitting a transfer function model via statistical methods. Matlab's System Identification toolbox is designed for this purpose.
I suspect your course professor did not specialize in control, and/or doesn't have real world experience. This is not unusual of course. Faculty are often assigned courses to teach, and some of these courses are outside their area of expertise.