I also read that Merriam-Webster article in the past (the second link you wrote) while I was writing my previous post in order to double-check and I believe the writer is wrong for the same reasons that I pointed out before. It's the lack of a comparison which makes it incorrect. Again, I still maintain that "more importantly" should be used neither as an adjective nor as a transition. By transition, I mean taking the focus from one subject to another.
If I could, I would even contact the writer of that M-W article to debate it further with them, and I considered it, but they did not leave a way to contact them (other than Facebook and Twitter, which I don't use) and there also isn't even a name attributed to that article. Actually, I think those Facebook and Twitter buttons are just for reposting it. They're not even contact methods.
I'll quote the relevant paragraph from my post once again,
The main point is that "more important" should be used for comparative purposes. It is used when something is previously mentioned and the speaker wants to then point out another matter which is of greater importance than the previously mentioned matter(s). When a sentence is started with only "importantly" there is no implicit comparison with a previously mentioned matter, thought, or idea. Using "importantly" without "more" to start a sentence just denotes that the words following "importantly" have been deemed important, but not necessarily more or less important than other matters. Again, when no comparison is being made, using importantly, by itself, is correct.
I think we'll have to agree to disagree, but I thought I'd at least respond. I already know it's a losing battle precisely because "more importantly" has come into widespread use, similar to how it's become common in English to say, for example, "there's a lot of people over there" when it should be "there
are a lot of people over there" since "there is" shouldn't be used with a plural noun.
I went off on a tangent, but that's the way I see it, at least. I've read local newspapers (I won't say which, because I don't want give away my location) that start off sentences using "More important" instead of "More importantly." I used to use "more importantly" myself, but reading it as "more important" in that newspaper is what got me to start thinking about it more and I came to the conclusion that if you're starting a sentence with "More importantly," then you're doing it wrong.
I'll finish off with my own link dump now (lol):
https://english.stackexchange.com/questions/229517/is-more-importantly-good-english
https://www.quickanddirtytips.com/education/grammar/important-or-importantly
https://english.stackexchange.com/questions/203917/most-important-vs-most-importantly
https://www.grammarphobia.com/blog/2019/01/important-importantly.html
https://guidetogrammar.org/grammar/grammarlogs3/grammarlogs422.htm
A relevant quote from the last link: "Many writers will use the adverb form, "importantly," thinking that the phrase modifies a verb in the main clause; usually, however, that is not the case. The phrase will almost invariably modify, adjectivally, the entire clause,
and the adjective form, "important," will suffice. Burchfield says that "more importantly" was a despised construction during the 1970s and 80s, but that nowadays both phrases seem be used about equally and with equal acceptability." [Authority: The New Fowler's Modern English Usage edited by
R.W. Burchfield. Clarendon Press: Oxford, England. 1996. Used with the permission of Oxford University Press.]
Note: This last quote (right above) helps support my earlier note about how "more importantly" has really gained more of a modern-day acceptance, when this was not always the case in the past, even as recently as the 1970s and 80s.
None of these links actually
settle the debate, but they are thought-provoking and good reads on the debate in question.