Most color processes until about 1932 were two-color. You could make an exception for screen plates like Autochromes which would reproduce all three primaries or complimentaries, but even du Hauron's famous 1877 print of Agen, France appears to be a two-color process. Most colors we see in nature tend to be muted, or "blends" of the primary colors of blue, green and red. Rarely do we see an object that is pure blue or pure red or pure green, (or yellow, magenta or cyan for that matter). So when photographing natural objects like human faces or a hillside or a cow the process doesn't have to express a pure yellow or red. Unfortunately people noticed that you never saw a picture of a sunflower or the blue sky or an emerald which looked right and there developed a push for "full color".
In 1922 Technicolor had a great success with a two-color film which starred Anna May Wong called "The Toll of the Sea". Since it was set in China the producers apparently felt that people would think the limited color palette was just because those colors were popular in China. A mixture of dyes was used for the red and a mixture was used for the green elements, US patent 1807805. Two-color sequences occur in "The Phantom of the Opera" 1925, "Ben Hur" 1925, early talkies of Laurel and Hardy and the Three Stooges, When sound came in there was a flurry of two-color films. Parts of some prints survive: "Gold Diggers of 1929", "Follow Through", and "The Mystery of the Wax Museum". Other companies besides Technicolor preferred to work in two-color because it was much simpler. Kodak even manufactured a film stock for release prints with emulsion on both sides to facilitate the methods of these independent processes.
What hindered the introduction of a three-color "full color" process was the development of a camera which would take three separation negatives. No special camera was required for animation so Disney's cartoon, "Flowers and Trees" 1932 was a three-color process, probably the first from Technicolor. Most color cameras worked as bi-packs: two strips of film were sandwiched face to face. The front strip was blue sensitive only and it was dyed yellow to act as a filter. So the front strip only photographed blue light and then the yellow filter allowed only red and green light to pass to the rear strip of film, which was panchromatic.
However, the use of acid dyes when these films were printed allowed much richer colors to be presented, even in the two-color processes. It sort of made up for the lack of full color.