Do working electrical engineers in circuit design ever use textbook formulas for rise time, peak time, settling time, etc
These calculations are absolutely used by professional EEs, for some on a daily basis. However, for many this job has been given to simulation software, such as LTSpice, which is also used on a daily basis. Generally the simulation is much faster to complete, so it is much more productive than doing the calculations by hand.

I generally use the formulas only to get a general idea of what to expect (say, within an order of magnitude), and leave the actual number crunching to the simulators.