The covariant phase space idea, in one sentence, is that there is a natural symplectic structure on the space of the classical trajectories of a system and that the usual $(q,p)$ coordinates just happen to be a convenient parametrization of this space. The advantage is that you don’t have to use this parametrization: there’s a construction of the phase space without any unnatural choices for field theory or for theories with higher derivatives.
I understand the pitch above. I’ve also seen several review articles 50+ pages long which describe the relevant (and unsurprisingly heavy) mathematics. However, I find myself lost reading them because I don’t know what to expect on the journey. Why should there be such an object? What are the crucial points leading to it? (And what’s the technical stuff I can skip on first reading—i. e. all the rest?)
Can anyone give a overview of this story that’s short enough that one can see all the logic at once?