Yes, projectors with 3LCD tech is what you are looking for. They produce all 3 colors at once via 3 distinct lcds inside the chassis and mix them ahead of time. There are a few to choose from, but they all cost above 3000.

The reason why projectors don't use a single rgb lcd (like monitors) to produce the color is the same why all sub 5000$ projectors use pixel shift to fake 4k resolution: Too much light is blocked by the lcd itself if the individual pixels become too small.