chore: prefer ArrayPartitionStyle to preserve nesting structure

[DhairyaLGandhi]

Checklist

• Appropriate tests were added
• Any code changes were done in a way that does not break public API
• All documentation related to code changes were updated
• The new code follows the
  contributor guidelines, in particular the SciML Style Guide and
  COLPRAC.
• Any new documentation only uses public API

Additional context

Ref #480

Add any other context about the problem here.

[ChrisRackauckas]

Rebase for the tests?

[DhairyaLGandhi]

I have a question, what purpose does nesting ARrayPartitions serve and is that a structure that should be maintained? Because we could always flatten the paritions of the children as partitions of the parent directly right?
The sizes would need to be consistent of course.

[kellertuer]

For the context in #480 originally comes from, no please do not flatten the structure: In that setting we handle elements on a product manifold with ArrayPartitions. We do not have to dive into manifolds, set products are enough:

Imagine you are on a set $A(n) = \mathbb R^n \times \mathcal P(n)$, i.e. the set of tuples where the first component is a vector and the second is a symmetric, positive definite matrix.
For the simplest case consider $B = A(2) \times A(3)$ so tuples where the first is the 2D case from before, the second the 3D case.
Surely any $b \in B$ should be an array partition of array partitions, i.e. have two entries, each of which is an array partition.

If you internally handle that in a fancy-flattened way, sure, feel free ;) But on the outer I would like to access the first or second element of b without having to thing about what these two elements are.