First, every odd prime divides some Mersenne number. (In fact, every odd integer divides infinitely many Mersenne numbers.) So that is no restriction at all.

Second, the claim from Wikipedia is correct, for the reasons given in CHUAKS answer. We can improve the Wikipedia claim to the following equivalence: A prime $p$ is Wieferich if and only if $p^2|(2^{{\rm ord}_2(p)}-1)$. Here ${\rm ord}_2(p)$ is the order of $2$ modulo $p$, and it is a divisor of $p-1$. (Generally, it is not necessarily prime.)

Third, Wieferich primes can potentially occur to odd exponents in $M_q$, as pointed out by Alexander Kalmynin, which is why your argument breaks.

First, every odd prime divides some Mersenne number. (In fact, every odd integer divides infinitely many Mersenne numbers.) So that is no restriction at all.

Second, the claim from Wikipedia is correct, for the reasons given in CHUAKS answer. We can improve the Wikipedia claim to the following equivalence: A prime $p$ is Wieferich if and only if $p^2|(2^{{\rm ord}_p(2)}-1)$. Here ${\rm ord}_p(2)$ is the order of $2$ modulo $p$, and it is a divisor of $p-1$. (Generally, it is not necessarily prime.)

Third, Wieferich primes can potentially occur to odd exponents in $M_q$, as pointed out by Alexander Kalmynin, which is why your argument breaks.

First, every odd prime divides some Mersenne number. (In fact, every odd integer divides infinitely many Mersenne numbers.) So that is no restriction at all.

Second, the claim from Wikipedia is correct, for the reasons given in CHUAKS answer. We can improve the Wikipedia claim to the following equivalence: A prime $p$ is Wieferich if and only if $p^2|(2^{{\rm ord}_2(p)}-1)$. Here ${\rm ord}_2(p)$ is the order of $2$ modulo $p$, and it is a divisor of $p-1$. (Generally, it is not necessarily prime.)

Third, Wieferich primes can occur to odd exponents in $M_q$, as pointed out by Alexander Kalmynin, which is why your argument breaks.

First, every odd prime divides some Mersenne number. (In fact, every odd integer divides infinitely many Mersenne numbers.) So that is no restriction at all.

Second, the claim from Wikipedia is correct, for the reasons given in CHUAKS answer. We can improve the Wikipedia claim to the following equivalence: A prime $p$ is Wieferich if and only if $p^2|(2^{{\rm ord}_2(p)}-1)$. Here ${\rm ord}_2(p)$ is the order of $2$ modulo $p$, and it is a divisor of $p-1$. (Generally, it is not necessarily prime.)

Third, Wieferich primes can potentially occur to odd exponents in $M_q$, as pointed out by Alexander Kalmynin, which is why your argument breaks.