What a great question. However, before I get to it, I want to give my qualifications (due to the comment graveyard below). Hopefully, this comment stays up.

I have a Ph.D. in Polymer Science in Engineering; my academic work centered on rubber chemistry, but I'm currently working as a Coatings chemist at a large corporation that focuses on manufacturing architectural materials. I'm also an amateur historian that focuses on 20th-century Military Aviation (used to be a flaired user here, but grad school got in the way of maintaining my subreddit activity).

Ok, on to the topic. I've interpreted your question from a technical standpoint, rather than the standpoint of a human psychologist or a marketing historian (not saying my interpretation is any better or worse than those). The short answer is that paint technology changed significantly as a result of environmental regulations brought on by the US Environmental Protection Agency (EPA). The EPA was established in 1970 to 1) address the growing public concern of the ways in which pollutants cause harm to the public and the environment 2) provide regulatory oversight and guidance to manufacturing entities/corporations/companies/individuals on how to lessen the environmental impact of chemicals and pollutants. Paint technology before 1970 was very basic, in terms of the number of ingredients that were required to make usable, stable paint. After 1970, a lot of these ingredients were banned due to their environmental toxicity, so paints suffered greatly in terms of longevity, stability (both environmental stability and color/mechanical stability), and aesthetic properties.

To give a more detailed, technical response, I want to first give a few definitions and some basic information on the subject.

1. Paints are more generally referred to as coatings. The primary job of a coating to provide longevity and stability to a substrate (i.e. the thing that you are putting the coating on). Coatings generally increase the service life of things like airplanes, ships, cars, buildings, etc by increasing the amount of time it takes for the substrate to be attacked by moisture/corrosion, UV energy from the sun, rocks and debris (think of cars rolling down the road and shooting rocks behind them) etc.
2. A secondary job of a coating is to provide some decorative or aesthetically pleasing covering to the substrate.
3. Coatings (paints) have 4 primary ingredients: binders, pigments, additives, and solvents. Binders are polymers, which are long chains of molecules that, when sufficiently large enough, act as a solid matrix that holds everything together after it dries/cures. Pigments give color to the coating. Solvents act as a liquid medium that either dissolves or holds all of the other ingredients in a gel-like suspension. Finally, additives are generally "all the other stuff" that is added to the formulation. We'll get into that later. They all work in concert to make the coating work and look aesthetically pleasing.
4. Polymer science, as an academic discipline, has really only been around since the late 1920's, early 1930's. Coatings and Paints, however, are unique in that they are literally among the oldest artifacts of the human species ( a quick Google search shows cave paintings going back 64,000 years).

Going back to pre-1970, manufacturers of metal-based goods (i.e. original equipment manufacturers, OEM) knew that in order to extend the service life of their piece of equipment, they had to coat it with something. And they had it easy. As I mentioned earlier, coating formulations back then were very basic, primarily due to 1 ingredient: Solvent. Solvents not only hold everything in suspension in the bucket, but they also affect coating properties like drying time of the coating, open time of the bucket, and most importantly: how the pigments/polymers orient themselves onto the substrate. For OEM manufacturers, they could buy some paint, put it on their equipment, and if it didn't look like what they wanted, they could tell the coatings chemist where they wanted improvements (higher gloss, better color stability, etc.), The chemist would go back into the lab, use a different solvent, then take it back to the OEM manufacturer. Rinse and repeat until the OEM manufacturer was happy. It was fast, and it was cheap.

When the EPA enters the scene in the 1970s, one of the first pieces of legislation that they pass is the Clean Air Act of 1970. This broad and far-reaching piece of legislation set in motion changes across multiple industries, from the production line to the R&D lab. Further amendments to the CAA in 1977 and 1990 began to place serious limitations on the types of solvents that could be used in paint formulations. Almost overnight (in technological terms), paint formulations went from being 5-8 ingredients to 12-20 ingredients, with most of the ingredients falling into the Additives list.

Generally speaking, binders and pigments generally agree with solvents, usually the more oily, the better. These organic solvents allowed fast dry times, great mixing power with all of the ingredients, but also allowed an easy way to tune how polymer and pigments oriented themselves during the drying process. If oriented just right, those pigments would POP and you would get some great color out of it. Due to environmental regulations, however, oily solvents (i.e. more petroleum based) became heavily restricted and started to be replaced with water. Water, while being a very environmentally friendly solvent, generally does not mix well with paint ingredients, does not evaporate quickly, and pretty much leaves a wrinkly mess of a coating rather than a smooth film. To overcome these limitations, additives have to be included into the paint formulation. Compounds such as drying aids, coalescents, surfactants, dispersants, thickeners, defoamers, and anti-foamers are just some of the additives that have to be added into a "waterborne" paint formula. Adding one thing usually causes an imbalance somewhere else, so you generally have to add something else. You're constantly playing catch-up to try to balance all of these properties together.

Also, it wasn't just the solvents and additives that changed; coatings chemists began experimenting with modifying the binders and pigments used in the formulations as well to try to make them more "water-friendly". Literally all four families of ingredients within paint experienced a chemical renaissance to avoid running afoul of the EPA.

Fast-forward to today, Paints and Coatings is still an ever-evolving industry. Newer regulations are being implemented throughout the world, and coating manufacturers have to continue evolving their formulations to meet these demands. However, the technology used to make waterborne paints has matured significantly since the 1970s, so we are able to formulate paints with colors approaching those that we saw pre-EPA, without nearly the same environmental impact.

I think its important to note, alongside the technological, economic, and psychological considerations, that colors have their own cycles of popularity. During the 1950s and early 1960s, both interior design and fashion saw an explosion of bright colors:

A page from an early 1950s Lustron prefab-home catalogue

An illustration from a late 1940s Lustron catalogue

Illustrations/ads from 1950s interior design catalogues/magazines

1956 "Spangles" candy advertisement

Far from the idealized, nostalgic image of the period as being full of pastels and pink and turquoise (man, I wrote a good answer about this I'd love to link, but I can't seem to find it), there was a lot of bright red, lemon yellow, forest green, etc. Living rooms and kitchens were full of bold colors, often placed right next to each other in a way that we would find clashy and horrible today.

Lots of things changed in the 1960s, and popular colors were part of it. While people did continue to wear bright colors, earth tones began to be more important - both because of the rising counterculture and the rising interest in Colonial Revival: mustard, brownish orange, dark brown and wood grain, avocado. And what do you see in cars of the late 1960s and 1970s, though not exclusively?

Muted colors, especially shades of brown and orange.

In the 1980s, people certainly continued to wear and use bright colors, but whites, off-whites, and greys became very common parts of the interior design palette, as well as glass and chrome - a space-age rebound from the historical/natural trend of the previous period. It's unsurprising that car colors would become similarly utilitarian.

The reasons behind this are many and varied, but the strongest influences come from paint technology, dealers, and pragmatism. Not necessarily culture. As a car enthusiast the pervasiveness of shades of grey has been well documented by automotive journalists.

Way back in the days of the BRIGHT colors of those pictures you're seeing, paint technology was different. Those days they used lacquer or enamel paint, which while vibrant, dulled with age, and wasn't actually as glossy as modern paint. They also weren't as durable as modern paint. But they were pervasive and the best tech they had at the time.

As we got into the 80s, metallic paints became affordable, and testing standards for paints came about from OEMs. Metallic paint gives that sparkle as you move around a vehicle. It looks premium, so customers loved it. Prior to this point, silver was just a dull grey. Clear coats also became common, which are transparent layers of paint added on top of the actual color. They also resisted UV rays more than the enamels & lacquers, meaning the car didn't lose its luster.

This explains how silver came into being. White became popular because it became the color of technology. "Czorni: White made inroads because of other industries using it to demonstrate the face of technology. A software and computer company led the way by making white a popular color for phones, radios and gadgets," (Finlay, Wards Auto). Want your car to look advanced? WHITE.

In addition to that, these duller colors became so popular because they were easier to sell. Dealers don't want to wait around for the handful of people that want a purple car, when they can sell a black, white, or silver car to just about anyone. Per the Wards interview, customers are willing to negotiate on color for a car. This quick-salability of "boring" colors also means the next owner can sell them more quickly, improving resale. But per a Forbes article (further references by the Consumer Reports article - all linked below), bright colors can have a better resale value - likely because those that want a bright color will wait and find that bright color. It may just take longer than a silver or white example. Forbes also says that the scarcity of the bright colors will drive up value, given that the used car market is bigger than the new car market.

Is this salability the reason that greys were more dominant during depressions/recessions? Potentially, though Consumer Reports also implies customers' attitudes of the time may have had an influence.

SOURCES:

Falsenthal, Slate

Finlay, Wards Auto (interview with BASF paint design head)

Quirk, Consumer Reports

Gorzelany, Forbes