As written, the question seems to have a problem (pointed out by user35593): If the powers of $b$ really give all of $1, 2, \dots, c-1$ mod $c$, then $c$ must be prime, and so certainly we do not get positive measure. (By the way, the term ``positive lower density'' seems more standard in this context.)

So let me answer a different question: Fix an integer $b$, and ask for positive integers $n$ where the powers of $b$ generate a cyclic subgroup of maximum possible size. In other words, we want integers $n$ for which the order of $b$ mod $n$ is the exponent of the group $(\mathbb{Z}/n\mathbb{Z})^{\times}$. Do we get positive lower density here?

If we restrict our universe to prime $n$, there's a well-known conjecture of Artin that the answer is usually yes. For instance, if $b=2$, then about 37.4% of *primes* $n$ should have the stated property.

What if we don't restrict to prime $n$? There's a beautiful theorem of Shuguang Li that for every fixed $b$, the answer is **no**. The lower density of such $n$ is always $0$.

On the other hand, for most $b$, the upper density (defined as before but with $\limsup$ replacing $\liminf$ is positive), at least if you believe GRH (a result of Li and Pomerance).

All of this is explained better than I have done here in this paper of Li and Pomerance:
https://math.dartmouth.edu/~carlp/PDF/primitiverootstoo.pdf