I assume we agree on the actual facts about the denominations -- some are more conservative than others, some are larger than others, and the theological family trees of the denominations is a matter of historical record. It's the terminology of "mainline", "fundamentalist", and maybe "modernist" that's under discussion.
"Fundamentalist" and "modernist" were at one time used to describe the conservative and liberal sides in the 20th century controversies. I haven't heard the word "modernist" used in ages, except as a historical reference, so I think that word is slipping out of usage. "Fundamentalist" used to mean "holding to the fundamentals of the faith", but it has drifted in meaning so that it's now mostly a perjorative, and it's also ceasing to be a useful theological term.
The terms I hear more frequently now are "conservative", "inerrantist", or "evangelical" for the denominations that maintain the inerrancy of the Bible, and "mainline" or "progressive" or "liberal" for those that do not.
The LCMS are not Fundamentalist, in the modern usage of the word. I am aware of the beauty of LCMS liturgy; I attended an LCMS church for a year in college, and they were the ones who introduced me to liturgical worship, a gift for which I am grateful. Because their statement of faith asserts the inerrancy of Scripture, I would categorize them as "conservative" rather than "mainline". The ELCA would be their mainline counterpart.
I have not personally been able to track down the first usages of the word "mainline", though I think some church historians have done that research. I can speak to how I've heard the word used, but that's about it. I think it's useful to have a neutral term to refer to the Protestant denominations that value church history and tradition but that also reject inerrancy, and to my ear "mainline" serves that purpose well.