Well, not quite. Some details are actually different, but apart from them there is a main difference:
1. Sort objects by one coordinate
2. Detect whether two objects are closer together along that coordinate than the sum of their radii
3. For all objects were this is true, sort by the second coordinate
4. Detect whether two objects are closer together along that coordinate than the sum of their radii
5. For all objects were this is true, sort by the third coordinate
6. Detect whether two objects are closer together along that coordinate than the sum of their radii
7. For all objects were this is true, initiate narrow phase
If you substitute 1. 3. or 5. (even only one of them) with some sphere approximation, you can no longer sort them, making it a n² algorithm average case, which isn't an improvement.
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Note that the broadphase detection currently only uses less than 15% of the run time of the collision system (according to the latest profiling I did)
We don't use an explicit AABB either, it's just an implicit result of the algorithm.