"If, as you say, common descent "rests squarely on a specific view of the origin of life", then only one of the above hypothesis can be true for common descent to have occured."
Bzzzzzzzt. Wrong. Bad logic, try again.
My point was that if you remove assumptions about the origin of life, then your assumptions don't require you to posit monophyly. Then all you have is the evidence. Incidentally, as the paper I referenced points out, the evidence doesn't require it, either, and actually speaks against it in many ways.
Again, without specific assumptions about the origin of life, there is no reason to assume monophyly. That is not the same thing as saying that only one set of assumptions will get you monophyly. The point is that if you don't make origin-of-life assumptions, you do not get monophyly from the data.
The link completely ignores ribosomal RNA, whcih, if monophyletic, proves common descent, regardless of what other genes are laterally transferred.