That OP and the answers that followed seem terribly overwrought as they struggled to answer using mostly verbage. IMO, the OP could be answered better and with less effort with a block diagram.

The OP of that quesition asked what happens if you vary one thing while holding other terms constant. That is an iappropriate question if the terms in the equation cited are related by other equations. You can't see that in a single equation, but you can see the multiple relationships in a block diagram such as below.

A block diagram allows one to simultaneously visualize multiple relationships, both transiently and in the steady state. For example, Q:What makes speed change? A:Torque unbalance.

You can also see in a block diagram what you can and cannot manipulate to change the steady state. (coefficients and external inputs, but not the initial value of states)

My question: Are modern engineering students expected to analyze things like motors without first learning about differential equations and Laplace transforms? If I answered a PF question with differential equations or a block diagram, would that likely be over the top and too hard to understand?

Having just completed engineering in December I can say that we did DE and Laplace xforms in 2nd year maths, and then retouched on them in classes when they were applicable.
We then covered block diagrams in 3rd year 2nd semester when we took Modern Control Systems. The prof wasn't one of the worse that I've had across 2 degrees and didn't' want to teach the course. So we got to electromechanics in our last semester our book explained things both ways. Similar to how you did it with block diagrams and then other way in words and equations.
I found it easier to understand the concepts with the equations myself.

I do think however there is an issue with schooling now where students get through without the understanding that they require. Like in electromech we had students that would change 100cm^{2} into .1m^{2} and other really basic stuff like that. Which becomes an issue as the class has to slow down, and the students who are actually understanding get affected.

I learned diffy Qs and laplace a year or two before i learned about block diagrams.
I don't think you can fully appreciate the block diagram approach without having some apriciation for the frequency domain. If you do have that appreciation, then yes I think using a block diagram would be great. Just make sure you explain what each element of the block diagram is. simply looking at equations will not tell you what is the plant, controller, amplifier, sensor, feedback, etc without some level of prior knowledge

edit: in addition if a student is working on motors for a class with any depth and does not understand differential equations, they are not being taught correctly IMO