Why are odd $n$ such that $2^n\equiv1\pmod{2n-1}$ so rare compared to even?

This cannot be an exact answer, but might give a clue: for an expression $f(n)=2^n-1$ the set of primes $p$, when seen as candidates for becoming primefactors of that expression, becomes "filtered" by the $\operatorname{ord}_2(p)$- function, which is a divisor of $\varphi(p)$ (Euler's totient) : if $\operatorname{ord}_2(p)$ divides $n$ then the prime $p$ becomes a primefactor of $f(n)$.

But many of that $\operatorname{ord}_2()$'s are even, so for odd $n$ there are few prime-factor candidates . For instance, the primefactors $3,5,11,...$ can only occur in $f(n)$ if $n$ is even, and only the primefactors $7,23,31,47,71,...$ have odd such orders . (A quick numerical check says, that about 29% of the odd primes up to some k have odd $\operatorname{ord}_2(p)$).
(...) It seems, there shall be a little more to say, but I've to look at it with more time.