Some people do, and erroneously conclude that "space is fake". This would be laughable, if not for the part where they then persuade their friends and family to feel deceived. Artificially coloring images without disclosing it clearly upfront in a highly visible manner, ideally with some sort of watermark on the image, is the sort of white lie that contributes to the public's growing distrust of authority.
Yes, I know the caption below the image usually disclaims it. But as you say, most people don't notice or know it until somebody points it out to them, and too often the somebody who points it out is somebody who's peddling nonsense of their own.
Then these detections are put together into a false-colour image so we can visualise it. We have three narrowband filters, so it makes obvious sense to map them to the three primary colours in the image.
It's not about using weird filters on the telescope to make "doctored" images that look impressive. It's about using weird filters on the telescope to collect the best science possible, and then using that data to make an image.