Yeah, but I'm interested in understanding things the other direction: if there is not some major class of problems that are best described by symplectic manifolds, then why care about Darboux's theorem at all? If there are, why isn't that front-and-center? This article conspicuously avoids motivating symplectic geometry _at all_, which is so frustrating. It mentions connections to subjects, but it doesn't mention why symplectic geometry is _necessary_, rather than sufficient, for these connections.
Is "any hamiltonian dynamics on any phase space of a physical system" not a major enough class of problems for you? This is one way mathematicians study physics. Are you asking for applications of symplectic geometry? If so, here, I've been reading a cluster of papers in this area for a few weeks now:
No, I'm plenty aware of the applications. My complaint is that this article hardly mentioned them! What's the point of an article explaining a theory without motivating why it exist?
Yes, too many mathematicians write about areas of mathematics that could be useful to others, without giving any motivation, or any hand-waving explanations, or the critical distinctions that make it a useful approach.
There is a barrier of communication that (most) mathematicians don't want to take any time to overcome. Often, the opening is enticing, then it's straight into lemmas, formal language, and citing of famous results by name. Their slight inclination to explain it to others dissipates in the first paragraph, then it's off to the races to impress their peers.
This is easy to see on Wikipedia - most mathematics articles are utterly useless to any non-mathematician who wants to get a general appreciation of an approach to see if it could shed some light on their problem.
In almost all cases, it is better to read domain generalists coming the other way into the higher mathematics.
The blunt reason is that this article was written for mathematics grad students who already know that symplectic manifolds are a hot topic via diffusion.
