This is not true.
It's really not, and you are perpetuating the idea that pigeons are these super dirty animals.
Parrot droppings, which are wet, are a lot worse than pigeon droppings if the pigeon is fed properly, which are semi-dry, intact spots 90% of the time. Even the larger morning poops, in my experience, pick up intact in a napkin without leaving anything behind.
I have a house pigeon, a female of the Old German Owl breed, who flies freely in the house and she is not a health hazard. When out, she is relatively potty-trained and will before anything else fly up to her designated spot to potty, a door above some newspapers, and poop before going anywhere else in the house. She will even hold it until she comes out of her cage and wait to use that spot if she can. She is very smart, extremely affectionate and loving and an all around perfect house pet. She does not bite or scream or chew furnishings like a parrot. She is like a cat that flies, though more affectionate than most cats I know, and still independent enough to entertain herself.
Homing pigeons are higher-energy than German Owls and not as ideally suited, but it can be done. I would nonetheless recommend a fancy breed instead as they are much more content to spend time in a cage and lounge around rather than want to fly all the time.
My bird does produce a lot of dander when molting and drops a lot of feathers at that time, but it has never been disruptive and only requires a few minutes a day to vacuum up and wipe down the cage. My house is pretty spotless.
I have also had an indoor ringneck dove and they are in every respect "pigeon, but a bit less." Not as big, but not as smart, not as messy, but not as affectionate. I find my pigeon more rewarding, but the doves are cute.
Pigeons do make nice house pets to the right owner, and if you can handle parrots, pigeons are much easier in every way.