Consider an inertial reference frame $I$, and a point object $K$ which is at rest with respect to $I$. $K$ is not at the origin of $I$.
Consider another reference frame $I’$, with the same origin as $I$, but rotating along a specified axis with angular velocity $\omega$ with respect to $I$.
From the reference frame $I’$, the object $K$ appears to be revolving around the origin of $I’$ with angular velocity $\omega$.
In the inertial frame $I$, no external force was needed on the body $K$ to describe its motion.
On the other hand, in the non-inertial frame $I’$, we need a pseudo-force to describe the body using Newtonian laws. Since in this frame, the body is revolving, so the required pseudo-force is centripetal.
However, in a uniformly rotating non-inertial frame, the effective pseudo-force should be centrifugal, instead of centripetal. So we have a contradiction.
I am sure I am missing something, but what?