The big three are not vertically integrated. Ford recently was talking about the industry outsources a lot of components. They tell somebody, e.g. Bosch, to make a seat motor, give specifications, then Bosch designs and manufactures it. That they might use that part across multiple models isn't vertical integration. This is also apparently why they can't do software updates like Tesla can -- Tesla IS vertically integrated in all of the ways that matter, unlike them.
As for the few companies today, a lot of that likely has to do with cars back then being a lot less complex than they are today. If you ever open the engine compartment of a 1930s era car, you'll see what I mean. Note also when most became defunct.
https://en.wikipedia.org/wiki/...
Much of this added complexity is regulatory -- in those older times, you didn't need catalytic converters for example.
A lot of those companies also produced cars that were either dogshit, butt ugly, or both (I.e. AMC Gremlin.) They were either facing a buyout or bankruptcy in many cases.