Mark Hansen wrote:

So what about the time when the left-seat pilot is not under the hood.
It sounds to me like they are making an exception for this case. Can
the safety pilot be PIC even while the left-seat pilot is not under
the hood? ... seems like a stretch to me.

To begin with, there is no safety pilot when the pilot flying is not under the
hood. So your "Can the safety pilot..." question doesn't make sense.

Aside from that, there is no requirement that the acting PIC is actually
manipulating the controls. Anyone aboard with the proper credentials can act as PIC.