Jut from my own common sense it seems like geometry is important for understanding the relationship between abstract things and concrete things. It's easily understandable that shapes are described by geometry and it seems obviously useful. The square footage of a house. The volume of a bath.
If you try to describe what Calculus is or does, it's abstractions of abstractions. Rates of change or 'angle of a curve for a certain values. I think it's hard for students to see this as something useful or even see how it's a description of the world that opens up ways of understanding it.
I don't see calculus as an abstraction of abstractions. The fundamental idea is completely geometric: "break the domain of a problem into a bunch of pieces that can be easily described and related (e.g. by physics) then put the pieces back together." Time is a first-class dimension. Abstractions only enter the picture when you want to separate the problem of picking a mesh from the problem of representing mesh elements.
Differential operators perform the task of "breaking into pieces" in a mesh-invariant way. Differential forms are mesh-invariant pieces. Integration is the mesh-invariant description of putting the pieces back together.
It's convenient that differential forms can be interpreted physically (by normalizing, associating with geometric elements, etc) but I'd hesitate to associate them with any single physical interpretation (e.g. rates of change) because doing so de-emphasizes the generality of the approach; you can have a rate with respect to distance, area, or volume just as easily as a rate with respect to time.
Leibniz notation makes the hop from the geometric approach to the "operator that maps a function to a function" approach seamless, and since the latter description isn't nearly so intuitive, I've always suspected that the geometric approach could profitably be taught first.
If you try to describe what Calculus is or does, it's abstractions of abstractions. Rates of change or 'angle of a curve for a certain values. I think it's hard for students to see this as something useful or even see how it's a description of the world that opens up ways of understanding it.