In my uni Classical Mechanics course was a pre-requisite to QM to ensure that students have a good intuition about Lagrangian and Hamiltonian formalisms, because those are non-trivial concepts by themselves.
You can solve any problem by using only the Lagrangian, there is really no need for the Hamiltonian, which also has the disadvantage of not being relativistically invariant, like the Lagrangian.
Also the name of "Hamiltonian" is somewhat misused. The most important contribution of Hamilton has been the definition of Hamilton's integral, i.e. the integral over time of the Lagrangian. That is an extremely important function and it would have deserved better the name of "Hamiltonian", than the less important Hamiltonian, which also was not introduced for the first time by Hamilton.
How to transform the system of equations of Lagrange to the "Hamiltonian" form had already been described by Poisson, and then by Cauchy, the latter using a form exactly equivalent to that presented later by Hamilton.
The notation H for the Hamiltonian has nothing to do with the name of Hamilton. Lagrange had used H for this quantity in 1811, without giving any meaning to the letter, then Hamilton in 1834 has reused the notations of Lagrange, adding "function S" for Hamilton's integral, also without giving any meanings to the letters.