We probably can't "do the same again", so much of the Industrial Revolution (from my reading at least) was started with the huge amounts of wood, then easily accessible coal, then "spending" coal to get access to deeper coal.
If you started from scratch, there isn't really any easily accessible coal left.
And consequently, those surviving and rebuilding are going to have the residue of a civilization that supported many more people to work with.
Cast iron might be relatively rare, but would it be relatively rare for 1/1000th as many of us?
If you preserve facts, you can reboot pretty quickly, relative to the initial time they took to discover.
Imagine how quickly you could make high quality steel if you could skip all the fumbling and straight to the proper carbon mix! https://acoup.blog/tag/iron/
But it's not a panacea. A "cold start" of a complex system isn't just about replicating the system itself, it's about replicating the conditions that gave rise to it. (Witness how many ex-Googlers have founded search startups and failed to get traction, or how many people have replicated Facebook and Reddit clones and failed to get traction.) A lot of our industrial processes (like building microchips, photovoltaics, plastics, modern manufacturing) require a large base of tooling that itself requires prerequisites which may not be available. In some cases the raw materials (eg. crude oil, rare earths) are no longer accessible.
The knowledge is useful, but it'll be useful in the sense that then people can look at their current situation and bring portions of the past into it. That's probably going to involve a lot of scavenging and looting, because why dig iron ore out of the ground and fire it with coal dug out of the ground when there are large supplies of abandoned scrap steel in the world?
Which is where the point about population comes into play. Refining low-density elements (e.g. rare earth) into purified form is most of the difficulty and industrial base.
The available recyclable material from a civilization with a couple+ orders of magnitude larger population would be more than sufficient, even if chemical reprocessing is required.
Consequently, you run into an either/or. Either (1) there wasn't substantial loss of life (and industrial base), in which case no need to reboot, or (2) there was substantial loss of life, in which case there are more than sufficient recyclables.
And it's also hard to imagine a scenario by which we lack essential inputs. We've got more than enough crude oil or rare earths for the future. The only arguable example I can think of is helium...
True, but we also know the usefulness of what is beyond certain points so we don't have to do the stochastic walk through the economics to get there.
We know we want the printing press. We know we want glass. We know we want antibiotics. We know we want steam engines. We know we want electricity. We know we would want steel. etc.
Since we know that steam engines are useful, we don't have to wait for knowledge to get good enough and materials to get good enough to make them economically viable so we can gain the knowledge that putting them to general use is valuable. see: The Cotton Gin--anybody could have made it, but until the demand for cotton was sufficient there was no reason to take the risk.
Avoiding the technologic stochastic walk would be a big deal.
I don't think this is strictly speaking true - certainly not for the US. I believe the majority of US coal production (according to wikipedia at any rate) is surface level mining, not the traditional underground mines people think about. I know that's true for parts of the Appalachian basin, I'm unclear as to whether that's true for the Wyoming mines.
Europe might be in trouble, I believe the only coal readily available on the continent is "brown" coal (lignite) which is suitable for power production, but has too many impurities to be used for steel production.
In any case charcoal can be used as a substitute.
Which means starting from scratch would require a different fuel like oil, but that’s even harder to extract these days, let alone in a post-apocalyptic environment.
Metals are often in refined form, which means in many cases higher melting points. (E.g. pig iron is 1500K, steel is 2800K)
We're not even mentioning electronics, because the vast majority of it isn't weather resistant, which means your "left lying around" is gone pretty quickly.
Plastic is in many instances only reusable in its exact shape. Alkaline batteries last 5-10 years, so good luck with those. Solar cells, in the best case, 25-30 years.
But all of that doesn't really matter. You'll spend the bunch of your time trying to just secure water, food, and shelter. Every day you don't get started on fixing things is decay. Every day you don't spend on food is hunger. (Subsistence farming is back-breaking, never-ending labor)
And so it goes.
Subsistence farming without machines is back-breaking, never-ending labor.
The whole idea is therefore to get machines up and running again as fast as possible.
And it all depends on the doomsday scenario. In most cases, there should be enough machines left to scavange. Or after a while, enough animals to be hunted.
Potential biggest hurdle are social dynamics. Confrontation instead of cooperation. And then the last capable electrician in the are gets shot, because some other scavenger wanted to get his corned beef.
Animal hunting is mostly a settler fantasy. There's not a single place in the US that has sufficient animal population to sustain human nutrition for more than a couple hundred people. You will require careful husbanding. And, for anything larger than feeding ~100 people, you'll require feedstuff. Which you transport... Ah. There's the lack of energy sources again.
Social dynamics are the least of your problem: The confrontation fans tend to die out quickly, or secure a fiefdom within which they ensure collaboration. Human beings pretty much default to tribal behavior. And they favor collaboration even across tribes. (I recommend reading Rebecca Solnit's "Paradise built in hell")
Really, it's a tougher row to hoe than you think.
And while farming may seem like back-breaking labor to the Aeron chair set, it’s really not that bad. You are tied to the land, but in the scenario that’s under discussion a Disney cruise vacation isn’t in the cards anyhow. You’ll want to have draft animals though of course as tractor replacements.
But yes, with draft animals you can make a go of it. It just won't leave you a lot of time to tinkertoy with the "leftover machines".
I don't think it's an unsurvivable scenario per se. I just think that the idea that you already have a starting point for an industrialized civilization is either quite naive, or takes a quite liberal view of "starting point".