Relationship between the Galilei Group and the Phase Space

This question is kind of a follow up question to my last question on the need for canonical commutation relations and conjugate observables. A comment from Valter Moretti suggested that, given a position operator, we need a momentum operator that satisfies the canonical commutation relation in order to be able to represent the Galilei-group. This makes sense to me (I demand the theory not only to to give information about the position of the particle, but I'd as well like to translate it in space, rotate it, or put it into another frame of reference).

The next thought I had about that: If the Galilei-group is in that sense kind of "linked" to the observables of nonrelativistic QM, can we think of a similar reasoning for the phase space?

Let's say we know nothing about the lagrangian formalism, and motivate the hamiltonian formalism from scratch. I then always thought of the momentum simply as an additional number (additional to position) that we need to have enough knowledge about the system to determine it's future. Requiring the volume of a little cell spanned by position and momentum to be constant over time then leads to hamiltonian time evolution (and the introduction of the poisson bracket).

Can I think this differently and introduce momentum in classical mechanics the same way I can do it in quantum mechanics?

My thoughts are that I think I can't: While in quantum mechanics with choosing a formalism of operators and states I am already given a concept of observables being generators of transformations, this is not true for classical mechanics. Of course I can state that there should be a transformation that translates the position of a particle, but I don't see why this transformation should be linked to an observable in classical mechanics.

2. The Bargmann algebra in $$n$$ spacetime dimensions is a Poisson algebra with $$n(n\!+\!1)/2\!+\!1$$ generators $$H, p^i, J^{ij}, B^i, m, \qquad i,j~\in~\{1,2,\ldots, n\!-\!1\},\tag{1}$$ which act on $$n$$ spacetime coordinates $$t, x^i, \qquad i~\in~\{1,2,\ldots, n\!-\!1\},\tag{2}$$ in a canonical fashion. This in particular determines phase space.
3. In QM, the Poisson brackets $$\{\cdot,\cdot\}$$ get replaced with commutators $$\frac{1}{i\hbar}[\cdot,\cdot]$$, cf. the correspondence principle between classical & quantum mechanics.