You've incorrectly assumed that once we are at a tangent with the apex we have finished turning. In a 180 deg turn, once we are at a tangent with the apex, we still have 90 degrees to turn the kart through.
If both rear wheels were still loaded, regardless of slip angle of the tyres (which categorically isn't the rear tyres sliding over the surface), the kart would understeer because the inside rear would still be required to rotate at a speed different to the outside rear. Certainly, the speed difference is less than if the slip angle were zero, but the speed difference still exists and therefore the inside rear must remain unloaded.
If you have a slip angle then, by definition, the centreline of the kart is not identical to the direction of travel therefore the diff action exists.
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