Most importantly, we need to arrive before 3:00 p.m.
In my opinion, the correct phrase to use at the beginning of that sentence should be most important, and the same goes for more important. Both are really shortened versions of what’s most important or more important than that.
Without a doubt, more important and more importantly are used equally by various revered, reputable writers, so there’s no reason not to use importantly. Both forms are acceptable grammatically. I agree that they are both used by reputable writers; however, Most American grammarians tend to object to the adverb (importantly) and defer to the adjective (important).
According to Merriam Webster’s Dictionary of English Usage,
Objections are made primarily on grammatical grounds. More importantly modifies nothing in the sentence. But, from the same point of view, neither does Most important. So, a longer phrase is postulated (What is more important…) and ellipsis adduced to explain the inconvenient absence of “What is.”
What’s most importantly is that we arrive at 3:00 p.m.
This sentence doesn’t make sense. (It should be What’s most important is that… .) It’s like saying What’s most clearly to us is that she will start the program at 3:00 p.m. That is definitely wrong. Yet, we use importantly in the same way. Importantly means in an important way, just as slowly means in a slow way. Using the full phrase what’s more importantly seems to be wrong.
Ultimately, whichever way you decide to write it, make sure you use it consistently throughout your work.
Maybe most importantly is here to stay; but, I still think most important is the better way to go.