While looking at proofs that every finite integral domain is also a field (or merely a division ring according to Jacobson), I notice the arguments all start with something like the following.
given the ring is finite, there can only be finitely many powers $a, a^2, a^3, \dots$ before they repeat
This is certainly true, but it apparently neglects the additive group of the ring; one can just as easily argue there are finite many factors $a, 2a ,3a, \dots$ before they repeat. So, what does the "typical" element of a finite ring look like?
My bet would be a linear combination like $n_1a^{m_1} + n_2a^{m_2} + \cdots + n_ka^{m_k}$. If this is the case, however, it's not clear to me how the above argument works. So I must misunderstand one of the two.
Let's stick to commutative rings with unity.
The argument that the sequence $(na)$ repeats is not neglected. When $a=1_R$ we find that there is a least positive integer with $n1_R=0$. This is the characteristic of the ring, and we can regard $R$ as containing $\Bbb Z/n\Bbb Z$.
Your image of the typical ring as consisting of the polynomials in a fixed element $a$ is not accurate. There may be no such $a$ one can use. As an example consider the quotient ring $R=\Bbb Z/2\Bbb Z[X,Y]/(X^2,XY,Y^2)$. This consists of all elements $r+sx+ty$ where $r$, $s$, $t\in \Bbb Z/2\Bbb Z$ and $x^2=xy=y^2=0$. Then for any $a$ you take in $R$, there are at most $4$ elements of $R$ you can express as polynomials in $a$ with integer coefficients.