So, as I understand it, the invention of the cotton gin in 1793 was fairly directly responsible for calcifying slavery into the "peculiar institution" it ended up as shortly before the Civil War. Indeed, I've heard it said that without the cotton gin, slavery would have fizzled out in the US long before the Civil War-era.
This is based on a misunderstanding. Cotton wasn't the only viable use of slavery. It's just that, for most of the first half of the nineteenth century, it was the
most profitable use of slavery. Knock out cotton, and all that happens is that the slaveowners move on to the
next most profitable use of slavery.
In OTL, slaves in the USA were employed profitably in cotton, sugar, tobacco, rice, hemp, kinda-sort indigo, wheat and other small grains, cattle, mining, road, rail and canal construction, shipbuilding, blacksmithing, coopering, turpentine manufacture, brickmaking, ropemaking, and a whole host of other artisan and urban manufacturing pursuits. Outside of the USA, you can add a whole bunch of other more tropical crops (coffee, bananas, sisal, tropical fruit) to that mix.
So, what if? Let's say Eli Whitney hits his head right before he was supposed to invent the cotton gin, and for some reason no one else comes up with the idea until at least the mid-1800's (or whatever). What happens? Does slavery get phased out in the US around the time the British dropped it (early 1800's), or does it go on?
As an aside, the cotton gin is the kind of invention which is practical enough and necessary enough that I'd struggle to see it delayed by so much as a decade. And if it is delayed somehow, you've just ripped the heart out of the Industrial Revolution, which was largely built around cotton textiles. But let's set that to one side for the sake of the WI.
What happens in the short term is a continuation of the same trends which started in OTL after the end of the ARW and before the invention of the cotton gin. Slavery based on tobacco cultivation expands into the uplands of South Carolina, Georgia and so forth. Slave-based rice cultivation expands a little too, as does sugar in those regions where it's suitable. (Largely Florida and the Gulf Coast.) Indigo remains moribund. Slavery based on long-staple cotton (which did not need the cotton gin) expands into the regions where it's viable, albeit those regions are limited. Slavery based on wheat/small-grain and hemp cultivation expands into Kentucky, and southern Indiana and Illinois. The pressure to make the latter two slave states, which was considerable in OTL, will become much stronger ITTL, and likely succeed.
Longer term, the westward pace of Southern migration will be considerably slower. It was the profitability of cotton which allowed the westward pace to proceed as fast as it did (without much immigration), and things will be slower. Population density, and particularly urban density, will be higher; another effect of slavery was the relative lack of small towns and cities. Slaves will still be used in agriculture: tobacco, long-staple cotton, rice, sugar, wheat/small grains, hemp, and so forth, but not as extensively as with short-staple cotton. Other uses of slavery (artisans, urban manufacturing, timber, construction, etc) will also be significant.
Also, is indigo plausible as a replacement for cotton? I know that it was successfully introduced into SC in the mid-18th century, but are there any factors that would limit its expansion throughout the south?
Indigo is not remotely possible. In OTL, indigo production collapsed after the American Revolutionary Wars, and never meaningfully recovered. It was always less profitable in the mainland USA, and relied on British subsidies for meaningful cultivation. After the ARW, that was gone, and it was mostly grown as an off-season crop to keep rice-growing slaves busy for the rest of the year. It was moribund long before 1860, and had no real potential to expand.