Why does the criterion for convergence of a power series not imply every series with bounded terms converges?

The fact that $c_n r^n$ is bounded, means that $r$ is in the set we're taking the sup of, so $r \le R$. But the convergence of $\sum_{n=0}^\infty c_n s^n$ is only guaranteed for $s < R$. It could very well be that $r=R$; a simple example is $c_n = (-1)^n$ and $r=1$.

The condition $r<R$ is not equivalent to $c_nr^n$ being bounded. It is possible that $c_nr^n$ is bounded for $r=R$ as well, and in that case we cannot conclude that the series converges.

The problem comes in the last step. Just because $\sum_{n=1}^\infty c_nr^n$ converges with $r \lt R$ you cannot conclude that $\sum_{n=1}^\infty c_nR^n$ converges. As an example, let $c_n=1$ for all $n$. We note that $R=1$ here. $c_nR^n=1$, so is bounded. For any $r \lt 1$, $\sum_{n=1}^\infty c_nr^n$ converges absolutely, but $\sum_{n=1}^\infty c_nR^n$ does not converge.