I've read in a bunch of different places over the years that we couldn't build a Saturn V today if we wanted to, because the tooling was scrapped and the blueprints shredded. What was the government's motivation for doing this? Why destroy knowledge that (I imagine) would've cost a trivial amount of money to keep stored somewhere in case it came in handy years down the line?
It's not uncommon to scrap the tooling involved in a specialty manufacturing project for cost and security reasons. There's plenty of examples, but it's most common in automotive and aviation circles. We tend to think of this as a wasteful process, but there are very good reasons to do so.
Think of what it takes to build a car. You're not building just one, so you need an assembly line. That means building a factory, a vast space to house the assembly line and the people who work on it. That's a major expense.
You'll need specialty tools, too — dies and stamps to create things like door panels. A lot of engineering goes into these door panels, a lot of time, manpower and labor is behind the design. You want to keep those engineering details secret — there's no sense in letting your competitor get a free manufacturing boost if you've created a superior door.
So you contract with a trustworthy tool-maker and ensure that you're the only company that can make this car door. Stamping a car door isn't a delicate process, so you've designed your factory to house this piece of equipment. You've installed a sturdy foundation, you've put in the correct electrical and other connections — the building is as much built for the tool as the tool is for the car.
You run the assembly line for 10 years, and the door-stamper pays for itself. It manufactures millions of doors without a hitch. Unfortunately, people's tastes have changed in 10 years, and it's time for a new car. You realize that instead of building a whole new factory for this new car, you can use your existing factory. If you're smart, you'll incorporate some of the parts from the first car into the new model — the chassis and transmission for example, are still reliable. The doors, on the other hand, will be changed.
That means designing a new door and a new tool to make it. What do you do with the old tool? It'd be almost impossible to remove from the factory intact — it's an enormous piece of equipment, that stamper, and it's practically built into the building. It's not a drill press or a lathe, something that can be repurposed. It's a cast-iron tool built specifically to stamp doors — your doors. Though it's 10 years old, you're concerned that someone might use it to make spare parts for your cars, and you need to be the only company doing that, or you'll lose money. Worse than that, your competitor might be the one doing that.
So you dismantle the door-stamper. You scrap it, because you know that recycling ensures you'll at least get some money out of the tool.
Keeping specialty manufacturing tools is darn expensive. If you want to preserve them, you have to preserve almost a whole assembly line, not just the tools themselves. Furthermore, it creates a security hazard. For something as important as a fighter plane or rocket, a foreign government is an even bigger threat than a competing car company.
Folks with an interest in aircraft or rockets tend to look at things like the destruction of the Avro Arrow's tooling or the SR-71's tooling in a vacuum. They don't think about the opportunity cost for keeping those tools. If you keep these tools (and the assembly line that goes with them), what can't you build as a result. Are you going to miss out on the next great project because you're holding on to old equipment? You've decided you're not going to build any more Deloreans, so you take the tooling and sink it to the bottom of the ocean so no one can copy your designs.
The short answer is they didn't destroy the plans and they would not be useful anyways.
I don't know the origin of this claim but this article [1] suggest it was from a book published in 1996 called Mining in the Sky. The same article states that Paul Shawcross, from NASA's Office of Inspector General said the Saturn 5 blueprints are held at the Marshall Space Flight Center on microfilm. So in fact the blueprints do exist.
However, they are useless for building a new rocket. The big problem is most of the parts are not manufactured anymore. For example the computers that flew the Saturn 5 use parts that haven't been produced in decades. Sure we have better and faster computers but that requires redesigning the control system. Similar problems happen throughout the design of the rocket. Really designing the rocket is not the hard or expensive part of building a rocket.(EDIT: This is not accurate as u/drobb points out. I should not of put it in in the first place) So if you are going to start building a new rocket you might as well redesign it.