I apologize in advance if this amounts to thread-jacking (and will accept any guidance from the mods on this), and I do understand that you are asking the question in earnest, but I don't believe that questions like these - made in isolation - are conducive to fully understanding the circumstances that our country faced during these crucial months. To be clear, it is appropriate to assess Lincoln's statecraft - or lack thereof - during these months. But I think exploring this question is only useful when examining the other side of the coin, so to speak. What could leaders in the South (Confederacy or the Confederacy-leaning states) done better or differently to avert the situation? Rarely do I see this side of the question get raised.
In November 1860, Lincoln won the election. No allegations of "rigged system" or fraudulent result (except perhaps that Lincoln was not even on most ballots in the South). Yet, when did leaders of the South ever indicate that they accepted the results of that election and would try to work within the system as the minority party? Lincoln had made it clear that he was not going to challenge slavery within each individual state. Yet, the Cotton States seceded before Lincoln was even inaugurated. What could they have done differently? Could they have pledged their loyal support to Lincoln and then tried to work out compromises about slavery in the territories (or employed delaying tactics until the next election)? Could they have taken note that most of Europe had abolished slavery and so taken the lead on proposing some pathway for abolition in the States (e.g., compensated emancipation)? Could they have made some form of motion within Congress for multilateral secession - an agreed parting of the ways among the States? Of course they could have, but didn't.
Respectfully, questions like the one you pose (and again, I understand you are raising it in good faith) seem to me to hold Lincoln to a higher standard than Southern leaders; i.e., it's okay for Southern leaders to disregard the election and attempt to form their own country, while Lincoln - who just won the election based on a clearly articulated platform - somehow has to act like he lost and compromise on that platform as a condition of peace, unity, etc., and without any guarantee that South will engage as a loyal minority? This seems to me to be a "heads I win, tails you lose type of scenario." So, while there may be some things he could have/should have done better, let's not forget that Lincoln was reacting to conscious decision by at least the Cotton States to separate from the Union on the basis of losing a democratically-held election.