Efficient decoding of irreducible binary Goppa codes and the role of matrix P in McEliece cryptosystem
If one knows $\gamma_1,\cdots,\gamma_n$ and $g(x)$ then for any $n$-long binary vector $V$ one can form the polynomial
$$v(x):=\prod_{i:V\cdot e_i} (x-\gamma_i)$$
and the syndrome polynomial $s(x)=v'(x)/v(x)\pmod{g(x)}$. This is all the information required by Paterson's decoding algorithm, independent of the basis used for the generator matrix and/or parity check matrix.
Section 3.1 should be read as an argument of the importance of keeping $M$ and $P$ secret as they are essentially the private key. It should be a very unreasonable assumption that the attacker can obtain these as this would break the system.
The recovery of $g(x)$ starts off as pretty much as you describe. For $V$ any codeword, compute the corresponding $v'(x)$ which we know is a multiple of $g(x)$. If we do this for two codewords, we can take the GCD which must also be a multiple of $g(x)$. After a handful of GCDs we have almost certainly recovered $g(x)$.
Note that that there is a equivalence between Goppa codes where columns are relabelled according to a linear function and $g(x)$ undergoes the corresponding linear change of variable. Recovering the columns labels of any equivalent code would also allow decoding.