Well, yes and no.
You are correct that Lincoln and his cabinet were careful to NOT make the war about slavery in the early stages. It was about preserving the Union. And though anti-slavery himself, Lincoln was not about to risk the Union to emancipate slaves. He figured it would burn itself out eventually. But was not about to risk the Union to free anyone.
BUT... the roots of the Civil War were based on slavery. It was the basis of the Southern economy. The 'wealth' of the small percentage of Southerners who owned the vast majority of slaves was based on... slaves as capital. The value of slaves added up to more than the entire value of all the railroads and Industrial companies in the North. And Southern Agriculture, heavily driven by cotton to feed the mills of the North and of England... was totally based on slavery. Sugar, too. Plus there were fears that 'emancipation' would result in massive bloodshed or strife... what were you going to 'do' with slaves?
Then there was the political angle. Up until the 1850's, there was a careful balance in Congress of Slave vs. Non-slave states. Remember, each 'slave' was 3/5ths of a (non-voting) person in allocating congressmen. So Southern plantation owners were loath to allow anything change that balance and strip their power. They felt the Northerners would crush them politically if the balance were tipped.
So when new states were being added to the Union (Kansas and Missouri and California, etc.) big fights were coming up over slavery's spread. It was not a moral issue of 'do we own slaves or not.' It was all about 'do we keep our power in the Congress!
Add in the toxic effect of the abolition movement... the nutballs of the Nineteenth Century. These folks ranged from the "Meh, I don't like this" to the utter murderous zealots like John Brown (who hacked up people with swords and fomented actual revolution." These zealots were like BLM or "Occupy" today. And they turned the Southern Militia systems from sort of a joke... a local society phenomena... into a serious movement of arming aganist rising slaves, abolitionists and preparing for succession.
On the other side, you had radical secessionists. Again driven by various causes, but largely to 'protect the Southern livestyle' which was utterly dependant on maintaining slavery.
This was what led to the events of late 1860/early 1861. And while noone 'said' slavery up front when the war started... the whole thing was most definitely rooted in slavery and the ripples it caused through society at basically every level. With the flames fanned by the new availability of "cheap newspapers" and pamphlets... the 1850's version of the Internet. Along with more leisure time brought on by wealth and industrialization (and slavery) that let people engage in 'causes' for the first time in America. Causes like secession and abolition.
Lincoln absolutely kept the war from being about 'slavery' for some two years. I think it was after victory at... Antietam? That he felt he had enough momentun to change the narrative to slavery. Pushed hard by rich Boston and NY Abolitionists and Frederick Douglas and his (white) followers. Who were all endless pains in the ass when it came to waging the war.
And, as you point out, as soon as the narrative shifted, there was a massive outburst of "pissed off" in the North. Ranging from the NY Draft Riots to whole units deserting the Union Army. Political fallout. Huge anger in the Irish community in particular (they were the 'low wage' outcasts in Northern Society -- and felt freed slaves would lower their wages and compete for jobs.) Sound familiar? So the shift from "Preserving the Union" to "Emancipation" was not an easy or popular one. And in hindsight caused a lot of issues we still feel today.
But to say the Civil War was not about slavery is only part of the story. It was 100 percent 'based' on the fact that America was founded with Slavery still permitted. It was the 'unfinished business' of the Constitutional Convention. And the compromises that allowed it to stand were destined to cause 'issues' in the future. They did. A shooting war that killed more Americans, if my numbers are right, than all our wars put together. (That may have changed.) Was it about slavery? Not if you read the period history prior to the Emancipation Procolamation.
But that's not the whole story. It was about slavery. Even if it wasn't.
Pardon the rant. Er... diatribe. Er... history blathering. Whatever. My coffee isn't working yet. Excuse for controversial remarks.
Sirhr