Confusion Between Dual Capacitor/Inductor Circuits
but why is this constant (step) voltage source apparently causing a
constant change in magnetic flux that causes the finite change in
current?

You seem to be changing the way you express things when you say this so I'm going to clear this up first. You appear to be equating: -
$$\boxed{\text{a constant change in magnetic flux WITH a finite change in current}}$$
In truth, any change in flux is also a change in current. The two are inseparable; one follows the other. I just wanted to clear that up. You probably meant what I re-phrased anyway!
From a comment: -

So does the emf approximately all drop across the inductor because the
inductor is what experiences most of the mag field change?

That is correct and what confuses some people is that if the back emf exactly matches the forward applied voltage, then how come there can be any current. Maybe that's what is confusing you. If that is the case, it can be answered but needs a little thought. Basically it boils down to this: -
$$\boxed{\text{back-emf tells us nothing about the current that flows in an inductor}}$$

One more thing, you say this: -

My understanding is that, assuming an LTI inductor, it is defined by
the relationship ϕ=Li

This is not quite true. This is correct: -
$$\Phi = \dfrac{L}{N}\cdot i$$
In other words, inductance per turn = \$\Phi\cdot i\$.
Hope this helps.