If we need to attract students' attention to Control theory and explain that dynamic systems are everywhere, what is more useful nowadays, CT or DT? If you teach both, what is your personal experience in the student's appreciation of examples?
Both are useful.
Personally, speaking I think the ideas are easier to convey in the continuous-time setting. For instance, the Bode plots are simpler to plot. Note also that many physical systems are continuous-time by nature. So, it makes sense to consider continuous-time systems first. Also, only the Laplace transform needs to be known.
However, controllers are implemented digitally now, which motivates to consider discrete-time systems. It requires a more advanced knowledge in signal processing than the continuous-time case. So, to me it is natural that it comes after.
However, while discretization is easy in the linear time-invariant case, this is much more complicated in the nonlinear setting or the time-varying setting. So, discretizing the whole control loop may not be easy.
Another issue with discretization is that you lose the inter-sample behavior. In other words, you do not have much information about what is happening between the samples. This may be an issue when one wants to find an optimal controller. Indeed, the discrete-time optimal controller obtained after discretization of the whole control loop may not be optimal anymore when looking at the continuous-time trajectory of the actual system in continuous time.
Finally, continuous-time disturbances may also be difficult to handle if they vary fast with respect to the sampling period. This may be a problem in the $H_2$ or the $H_\infty$ control of continuous-time processes using discrete-time controllers.
A workaround to all those problems is the consideration of a hybrid system that will consider the problem directly. Indeed, a control loop consisting of a continuous-time system and a discrete-time controller is hybrid by nature. This framework does not require any reformulation or transformation. It requires some knowledge about both continuous-time and discrete-time systems, though.
There are processes which are discrete-time in nature and those are computer systems or any systems where decisions or updates are made at specific instants. For them, it is perfectly natural to consider discrete-time controllers.