We’re not saying TVs are bad, not at all. But the larger the display, the more distance signals need to cover across the panel, and sheer physics mean latency and lag thus increase. Advances in TV CPUs mean that now there’s very little difference between a 55” TV and a 75” TV, but it’s still there.
That’s a disadvantage of internal displays, while projectors are external. To the projector’s processing components, it matters little if you have an 80” image or a 120” one. Total latency will be the same.
Screen size with TVs means the space a 4K screen’s 8.3 million pixels/transistors cover, so of course if that space is bigger, data takes longer to reach every pixel. That’s not an issue for projectors, as the processing is done in the projector, and the projected image is simply light, and not directly generated on transistors or pixels.
Another aspect of this to consider is resolution, which affects TVs and projectors in a similar fashion. The higher the resolution, the lower the framerate due to processing burden in the gaming console (or PC). Combined with screen size and other factors, higher resolutions result in high total input lag or latency.
Modern, high quality 4K gaming projectors give you 4K 60Hz, which is more than fine for PS5 and Xbox Series X, at just over 16ms of total input lag. That’s very competitive with the quickest TVs, and faster than most.