Why is the symplectic manifold version of Hamiltonian mechanics used in Newtonian mechanics?
The symplectic framework paves the way to classical mechanics on a Poisson manifold (i.e., with a Poisson bracket). See the book Mechanics and Symmetry by Marsden & Ratiu. https://cdn.preterhuman.net/texts/science_and_technology/physics/Introduction%20to%20Mechanics%20and%20Symmetry.pdf
The Poisson point of view unifies many classical systems that cannot be seen otherwise in a uniform way. For example, many fluid flow equations have a Hamiltonian description only when viewed on an infinite-dimensional Poisson manifold. See the survey article
PJ Morrison, Hamiltonian description of the ideal fluid, Reviews of Modern Physics 70 (1998), 467. http://www.ph.utexas.edu/~morrison/98RMP_morrison.pdf
and the book
Beris and Edwards, Thermodynamics of flowing systems.
The action principle, stemming from Fermat and Maupartuis, and developed by Lagrange tells us that a classical system extremizes the action $S~=~\int L(q,{\dot q})dt$ so that the following equation is obeyed $$ \frac{d}{dt}\frac{\partial L}{\partial{\dot q}}~-~\frac{\partial L}{\partial q}~=~0. $$ This defines a differential equation for each Lagrangian $L(q,{\dot q})$, which is equivalent to Newton’s second law. The action is a bare action plus a Hamiltonian under a Legendre transformation $Ldt~=~pdq~-~Hdt$, where this tells us that the Lagrangian is a form of constraint which sets the dynamics on some manifold of solution. This manifold is called a contact manifold.
The Hamiltonian can be showed with several steps of mathematics to obey the equations $$ {\dot q}~=~\frac{\partial H}{\partial p},~{\dot p}~=~-\frac{\partial H}{\partial q}, $$ which are the Hamiltonian dynamical equations. In a general setting the evolution of quantity $z$ is given by ${\dot z}~=~\{z,~H\}$, which is a Poisson bracket. The antisymmetric structure of this theory can be encoded according to Hamiltonian flows determined by a symplectic matrix $\Omega_{ij}$ which is antisymmetric. If we let $z~=~(q_1,\dots, q_n,~p_1,\dots,p_n)$ be a $2n$ vector, then for $\Omega_{ij}$ a $2n\times 2n$ matrix $dz_i/dt~=~\Omega_{ij}z_j$ is a general form of dynamics in symplectic form.
This is an outline of how this is developed. Arnold’s book on Math Methods of Classical Mech is a good source for a deeper study of this.
This question is somewhat of a physics history question about when and why. I dont have a specific reference as to who first introduced symplectic geometry into physics, but there are some broader indications as to what has happened.
The symplectic manifold idea is a formalisation (in terms of differential geometry) of the phase space idea. Looking up phase space in Wikipedia gives this useful reference and explanation:
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables.
A second factor that would have been relevant in this case was Liouville's Theorem. This theorem asserts that the density of points in phase space is conserved. It is about volumes in phase space, thus suggesting a differential geometry treatment of phase space.
Once it became clear (I dont know whether this was Gibbs or later) that the symplectic form exists as a geometric object similar in some ways to a metric, and closely connected with the Poisson bracket and the rest of classical mechanics, then it would be natural to describe the whole of classical mechanics in this geometric way.
This could be another topic that was first noticed by mathematicians, but only noticed by physicists after the advent of quantum mechanics (1920s or 1930s) when explicit connections between the two theories (Classical and Quantum) were searched for. Names associated with this are Weyl and Moyal (in 1949). More on these early phase-space to QM ideas in this Wikipedia link.
It also needs to be remembered as we discuss post General Relativity and post Quantum Mechanics eras, that phase space formulations of GR have been sought as an aid to quantization. All this is a highly geometric approach to Quantum Gravity, associated with many names although Ashtekar worked on this overlap. Eventually this led to his (co-) formulation of Loop Quantum Gravity.