No, not even close.
Alan T Nolan lists this as one of the components of the Lost Cause Myth in his essay "The Anatomy of the Myth", collected in the book The Myth of the Lost Cause and Civil War History (ed by Gary Gallagher and Nolan, Indiana University Press, 2000). McPherson says in Battle Cry that slavery was more firmly entrenched in 1860 than it had been in 1820. By 1860 the "stock" of enslaved persons was the single most valuable concentration of wealth in the US – more valuable than all the railroads and manufacturing concerns in the country PUT TOGETHER. The Mississippi History Now website has a page that says:
[C]otton was America’s leading export, and raw cotton was essential for
the economy of Europe. The cotton industry was one of the world’s
largest industries... In many respects, cotton’s financial and
political influence in the 19th century can be compared to that of the
oil industry in the early 21st century.
Groups don't typically just give up that much wealth/power, without an objection.
By 1860 it was illegal in the South for antislavery literature (including many Northern newspapers) to be sent in the mail: Southern inspectors were opening mail. (See for example here and here and here and here.) Free speech was infringed upon: it was illegal to voice antislavery opinions. By 1860 it was illegal in many Southern states to even BE a freed Negro: if you were an enslaved person and somehow were given your freedom, you'd better get the hell out of the state FAST. Kenneth Stampp writes in The Peculiar Institution (1956) that by 1860, many Southern courts had started reviewing wills where a dying slaveowner freed his enslaved persons, and overruling the will, ordering that the enslaved persons be kept in slavery. It seems that in the South, a slaveowner could do anything with his enslaved persons except free them. So the laws in the slave states were tighter by 1860 than they had ever been.
The Federal situation was even worse for enslaved persons (and freed blacks). The Fugitive Slave Law of 1850 was more draconian than anything in effect back in 1820: it was quasi-legal kidnapping. The 1857 Dredd Scott decision purported to make Negros ineligible for citizenship in any state – INCLUDING in the 5 states where blacks had held citizenship and been able to vote during the Constitional ratification era. In other words, the 1857 Supreme Court ruling took away black citizenship in 5 states where they had held citizenship since 1789. There was a very real fear in the North that the next ruling from Roger Taney's court could make it unconstitutional to EXCLUDE slavery from any state. Lincoln stressed this in his "House Divided" speech at Springfield in 1858, quoted on the Dred Scott wikipedia page. That same page contains a reference to Southern radicals boasting that the coming decade (after the Dred Scott decision) would see slave auctions on Boston Common. (Lemmon v. New York might have supplied the Court with an opportunity.)
And of course the slave states were looking to expand slavery into Mexico and Cuba. Many Southerners could foresee a paradise of slavery extending all around the Carribean basin. See here and here.
This census.gov link shows the actual number of enslaved persons growing from just under 700,000 in 1790, to just under 4 million in 1860:

That does not look like an institution that is dying out. It's an average growth rate of 2.5% per year, 28% per decade. The ending of the slave trade is not even a blip on this growth chart: 293,000 more slaves in 1810 over 1800, and 347,000 more enslaved persons a decade later. (And, of course the ending of the slave trade would not make a dent in this growth, once it was established. A few shiploads, or few dozen, is nothing compared to how many children 893,000 people can have.) The "delta" grew every decade, as you would expect with exponential growth. In 1860 there were 750,000 more enslaved persons than there had been in 1850, the highest delta ever. At that rate, there would have been over 5 million enslaved persons in 1870, an increase of 1.1 million over 1860.
Gail Jarvis and the Dunning school present an outrageously deceptive misinterpretation of some facts. First:
"Plantation owners and their families were well aware of the slave
revolts in Haiti and other Caribbean island where hundreds of Whites
had been slaughtered. These stories, coupled with reports of slave
uprisings in the American South, certainly must have conciliated many
a hardened pro-slavery stance."
Yes, Southern slaveowners were well aware of the revolts in Haiti and the Carribean, and reports of uprisings in the south. Stampp documents that in fact they were hyper-aware of them. That awareness created at atmosphere of suspicion and suppression. This did the exact opposite from "conciliating" hardened slavers. Instead, those reports induced periodic panics and reprisals, where slave patrols tortured and murdered enslaved persons who were suspected of plotting. This was an integral part of the regular pattern of terror and oppression that kept enslaved persons in line. The exact opposite of conciliation.
Southern awareness of Haiti et al extended to the decision to secede. Southerners told each other that Black Republicans would turn the South into another Haiti. (See Apostles of Disunion by Charles Dew, 2001) They said they had to secede to protect their families and especially their daughters.
Second:
"there were more than 250,000 Free Persons of Color in the South, not
only in major cities like Charleston and New Orleans but in smaller
towns."
250k free blacks in the South is a drop in the bucket compared with the 4 million enslaved persons: just 6% It is only one-third of the number of NEW enslaved persons in 1860 over 1850. The argument is absurd, maintaining that the 6% is "representative" while ignoring the other 94%.
It is a deliberate distortion.
Stampp also writes about the profitability of slavery. He says that slaveowners typically claimed that they lost money on enslaved children whom they fed and clothed thru childhood, and then later sold. Those slaveowners would deny that they were in the business of "farming slaves", which had negative connotations. They fed those enslaved children out of the goodness of their hearts, and lost money on the deal. But Stampp reviews the sale prices for those enslaved persons, as recorded in various contemporary publications, and concludes that raising & selling enslaved persons was big money for those plantation owners large enough to do it. He does not go into specific numbers; but this directly contradicts Ulrich Phillips' assertion. The profitability of plantation slavery was not tied only to the production of cotton, but to "farming slaves" also.
No, the idea that slavery was somehow on its way out in the antebellum USA is a complete fabrication by the Dunning school, without a shred of evidence supporting it. It's so at variance with the demonstrable facts that it has to be an outright lie, an attempt to deceive. And in fact, that's what the whole Dunning School of history was: propaganda to support & justify the Jim Crow system. Here is Eric Foner on it 10:
The traditional or Dunning School of Reconstruction was not just an
interpretation of history. It was part of the edifice of the Jim Crow
System. It was an explanation for and justification of taking the
right to vote away from black people on the grounds that they
completely abused it during Reconstruction. It was a justification for
the white South resisting outside efforts in changing race relation
because of the worry of having another Reconstruction.
All of the alleged horrors of Reconstruction helped to freeze the
minds of the white South in resistance to any change whatsoever. And
it was only after the Civil Rights revolution swept away the racist
underpinnings of that old view — i.e., that black people are incapable
of taking part in American democracy — that you could get a new view
of Reconstruction widely accepted. For a long time it was an
intellectual straitjacket for much of the white South, and historians
have a lot to answer for in helping to propagate a racist system in
this country.
It is interesting to speculate on when slavery would have ended, absent the civil war. Two possible "outer boundaries" for when plantation slavery would have ceased being profitable are the boll weevil infestation in the early 1920s, and the invention of the first really successful mechanical cotton picker or combine, the International Harvester model "H-10-H", in 1942. It's difficult to imagine slavery in America in the 20th century: those suggestions are just pure speculation about the absolute latest possible dates when plantation slavery might still have been economically viable. But of course enslaved persons could work in factories and mines, too.
Lincoln and the Republicans wanted to put slavery "in course of ultimate extinction", by limiting it to the existing slave states and keeping it out of the territories and any new states. The idea was that if slavery was kept where it was, then it would eventually die out, maybe with the aid of "compensated emancipation" where the government would pay the slavowner some discounted value for the freed slaves. I read somewhere that Lincoln's own estimate for how long it would take for that to happen was around 50 years. (I will update if I ever find my source for that.)
This of course is exactly what the slave states were afraid of, and why they seceded. Ironically, secession started the ball rolling that did in fact end slavery. But before secession slavery was still going very strong, with no end in sight.