In short, weird things happen when you combine things that don't combine in nature:
- the Earth as a perfect sphere
- the Earth spinning on an axis
Einstein's equivalence principle tells us that accelerations are all the same, no matter what's causing them. So you just add the acceleration vectors up.
The (real) Earth has an equatorial bulge because a stable surface can only exist where those vectors sum to zero:
- residual ("centrifugal") acceleration from the Earth's rotation
- the ground holding you up
This is called an equipotential surface and because of the centrifugal acceleration, on a spinning Earth this surface is an ellipsoid (well, mostly; there are small adjustments due to mass distribution).
The constraint of having a perfect sphere means that the materials making up
the Earth your fictional planet are strong enough to keep it from ballooning out and filling an equipotential surface.
So if you make your fictional planet a perfect sphere, what you're really doing is changing where #3 happens but leaving #1 and #2 the same.
In other words, you're creating a 26 km deep valley encircling the globe.
Water runs downhill.
So eventually, all of the water will flow down into the giant equatorial valley...