The simple answer to your question is that a recirculation loop is never a good substitute for a well-designed mixer, for at least two major reasons:

To accomplish the same volumetric pumping of a mixer, the pump would require about 10 times the power of the mixer.

Return flow from the recirculation loop creates an unpredictable and often short-circuit path through the vessel, therefore not providing good bulk uniformity in the vessel.
As a simple guide to the use of a recirculation loop, the loop should be designed for a specific purpose other than bulk mixing.

The loop may withdraw material from a poorly mixed region at the bottom of the tank, as from a stagnant region in the drain valve or from a pile of settled solids.

The loop may provide additional heat transfer through an external heat exchanger.

The loop may pass material through an inline, high-shear disperser to eliminate lumps.

The loop may pass through a static mixer for rapid addition of a reactive chemical.

The loop is essential for effective processing.
Other than a clear indication that effective mixing is not present in the vessel with the existing agitator, I cannot think of a good reason for using a recirculation loop as a substitute for an agitator.

It sounds as if you should attend a course on basic mixing.