Well, in the spirit of "food for thought", I'd say that one problem I have with the argument attributed to Mr. Reiner, is that he's only considering one aspect of the question, "What makes a continuing Church"?
Is it:
Reiner seems to be saying that if the Roman Catholic church can't prove their first claim (An unbroken succession of Leaders), then the Mormon claim to have "restored" the proper Leadership of the Church becomes the next, strongest alternative.
But I don't think that you can say that, until you first establish that an unbroken succession of Leaders is the key defining characteristic of a "continuing" Church, and that not only the Roman Catholics but that every Christian Church has lost such Succession (which the Eastern Orthodox, for example, might dispute on their own behalf; in addition, some Protestants -- such as some Anglicans -- do claim to be part of the Apostolic Succession of Bishops, just outside of Roman Communion). And speaking just as a Presbyterian, I might tend to place more weight on the latter two characteristics of a "continuing" Church, anyway.
So, there's a number of preliminary issues that need to be established as points of agreement or disagreement, before the main question being asked can even be answered. If'n that makes sense (hopefully I'm not rambling too much).
Thanks for listing the ideas of what constitutes a continuing Christian church. As far as the LDS church is concerned, and I'm fairly certain the Catholic church as well, it's only one thing. It's the "one aspect" of priesthood authority. Whether it is through apostolic succession, or that it was restored in the latter days directly from the original Apostle(s) during visitations.
No other restored church outside of the LDS claims to have had the priesthood restored in such a fashion.
While the LDS church declares that priesthood authority is necessary for valid baptism, the Catholic church accepts baptisms done outside of the church as long as they were done with a Trinitarian formula (in an emergency, they can even be performed by non-Christian as long as they are done "in the name of the Father, and the Son, and the Holy Spirit). Not being an expert on Catholicism (I'm learning), I suppose Catholics believe that the priesthood is necessary for the various rites and to offer the Eucharist.
As an aside and to tie this back into the original thread topic for minute, it does seem like Catholic priests do have more power readily available to perform exorcisms than say Evangelical preachers. But that is another topic.
Anyway, you're correct, it does matter what the definition of a continuing church is. Perhaps, that is what I was wanting you to tell me.
I'm open to opinions. Yet I tend to see the necessity of an unbroken chain of authority... Thanks!