I've updated this thread. It's almost ready for its final update. I'll post my reply here as a duplicate, for those who don't want to wade to the bottom of a long list of spoilers.
2013-07-29 The first replacement motherboard which arrived broken and that I shipped back to Asus arrived at Asus' loading dock at 09:53 this morning (local time at Asus' facility) and was signed for by "KFIELDS". All is well there.
Although I didn't receive the tracking information for cross-ship via e-mail, and hadn't heard from Asus since 2013-07-23 when I supplied them with mine, I wasn't frustrated by that, as I had good indication that I would be receiving a replacement.
I called Asus RMA today and didn't have to wait long on hold at all before I reached Paul in the RMA department. Paul was polite, helpful, and courteous and he was quite apologetic that he had to place me on hold to retrieve the information for me. He did verify the tracking number of the second (and I hope final) replacement motherboard, and I was pleasantly surprised to find that FedEx lists it as on a vehicle for delivery to me some time today as of 08:26 local time for me.
Which parent company owns which subsidiary is not germane to the discussion of this thread.
To the retail customer and to the customer seeking RMA, they're separate entities, as most customers will have no contact with PTSI and PTSI's involvement in the repair process of motherboards will be transparent to most customers. Any problems in the PTSI repairs area are not really for a customer to resolve, but rather for Asus to deal with internally.
The products I have bought were Asus (I owned two other, older, Asus boards which are still going strong in other people's PC's) and the corporate entity I dealt with on the telephone for RMA issues is Asus. The only time that PTSI got involved directly with me was when I was directed to forward pictures of the damage to Mr. Prada at PTSI. (I never did hear back from him at all, but it's obviously a moot point. I'm happy to let Asus handle that in-house.)