Why the NGBs are the parameters of the vacuum manifold?

I attended an Effective Field theory class where we studied the CCWZ formalism, in order to construct effective field theories for the NGB ( Nambu Goldstone Bosons), starting from a symmetry breaking pattern of a global group G into its subgroup H.

G is assumed to be a compact internal ( no space time ) connected semi-simple group.

In class we identified two types of generator the broken generators $X^a$ and the unbroken generators $T^a$, clearly the broken generators are related to the element of the coset $ \frac{G}{H} $, meanwhile the unbroken generators are related to the elements of the subgroup $ H $.

Now my professor said that since we have SSB we have a non trivial overlap between the particle state $ |\pi^a(p)>$ and an infinitesimal transformation along the direction given by the broken generator $ X^a $ ( this is just one of the results of goldstone theorem ).

Now here is the part that I do not understand, my professor said that from the results of Goldstone theorem, the parameters that describe an element of the coset space $ \frac{G}{H} $ should be "identified" with the goldstone bosons. I do not see a formal mathematical way to prove this to me, the only intuition that I got is that usually when we do SSB we can parametrize the field as

$ \phi(x) = \frac{(\phi_{0}+\rho)}{\sqrt2}exp\left({i\frac{\pi_{a}(x) X^a}{f}}\right) $

now if I do not look at the radial mode, (which in the context of EFT typically gets integrated out), I can see that the action of the NGB on the vacua is equivalent of the action of an element of the coset space namely:

$ exp\left({i\frac{\pi_{a} X^a}{f}}\right) \phi_0 $

since in general $\phi_0$ will stay invariant for elements of $H$ and near the identity I can always decompose the group element as product of an element of $H$ and an element of $G/H$

So yeah from this equivalence I can see why I can identify the two's quantity, provided that I promote the $\pi^a s parameter of the group to fields $\pi^a(x)$ giving them a space time dependence.

First question: Why this promotion (if its the right way of doing this) does not make the group G and its related symmetry from global to local ( like a gauge symmetry where the parameter of the gauge group depend on space time) ?

Second question/doubt: I checked the original paper of Coleman, Wess, Zumino ( the name is "Structure of Phenomenological Lagrangians", from 1969 ) here they write something that seems related to my problem i.e. " If we identify the fields of phenomenological theory with some particular set of coordinates on the manifold, the problem of finding all possible field transformation laws under a group is equivalent to finding all possible ways of realizing the group as transformations on a manifold."

Thus I do not know if this hints to the fact that all the possible values of $\pi^a(x)$ are just different realization ( as in numerical values ) of the parameters that parametrize the manifold.

I try to explain it better: For each different point of Minkowski space time $x$ we have set values of $\pi^a(x) s, thus I could make a bijection between this set and the set of all possible values of the parameters that parametrize the vacuum manifold ?

And does each set of values of $ {\pi^a} $ [ parameters of the vacuum manifold, which identify a point on the manifold ] relates the others by a change of coordinates on the manifold ?

Can this change of coordinate be seen as an "active" symmetry moving a point on the manifold ( one vacumm configuration) to another point ( equivalent energy but different vacumm configuraiton) ?

Lastly :

I presume that the manifold they referred in the paper is the vacuum manifold , and intuitively I think that if each point of the vacuum manifold is a different vacuum configuration then acting with elements of $G/H$ makes us go from one point of the manifold to another, thus starting from the origin every point on the manifold can be "described" by the same set of values that describe the element of $G/H$ that acts on $\phi_0$ in this sense the parameters of the coset space $G/H$ ( equivalently the NGB) parametrize the vacuum manifold. is this correct?

Sorry if this sounds confused or if I repeated myself, but I am confused,

if any smart person could help a silly goose like me, it would be much appreciated!

Thanks,

Good evening