We’re losing sight of what the main issue of debate was: Who gets to designate the PIC? That led to the definition of PIC, which in turn led to the definition of “operator, of an aircraft”. That’s because the latter designates the former.
Let’s assume that it’s rock solid true that flying training, or at least flying training conducted during a flight review, can only be conducted by the holder of a Part 141 certificate. I do not think it follows, from that assumption, that the holder of the Part 141 certificate thereby becomes the operator of the aircraft as defined. Para (a) of the definition of "operator of an aircraft" is about whether a certificate "authorises the operation" of the aircraft.
Take the example of an aircraft the registered owner and registered operator of which is pilot Fred. Fred flies over to aerodrome X where the holder of a Part 141 certificate “Part 141 R US Pty Ltd” are located, because Fred contacted them and they agreed to arrange for the conduct of Fred’s AFR. Instructor Joe conducts Fred’s AFR. Does that set of circumstances mean that Part 141 R US Pty Ltd became the operator of Fred’s aircraft for the purposes of that definition in CASR, during the AFR? If so, there’s waaaay more in the regs about the implications of being the operator of an aircraft, beyond mere designation of PIC, that Part 141 R US Pty Ltd needs to be worried about.
Set aside all of the distractions about whether an instructor or Part 141 organisation would “accept” a situation in which the owner could designate. If the Part 141 organisation becomes the operator of Fred's aircraft, merely as a consequence of conducting the Fred's AFR in Fred’s aircraft, Fred doesn’t get to choose who's PIC because - on this logic - Fred's not the operator and, therefore, Fred does not have the power of designation. But it also follows that the Part 141 organisation doesn't get to choose which of the operator obligations in the regs are complied with or not.