Sorry, but I think your post is a bit misleading to potential pet bird owners.
Parrots are captive bred to be pets. They are not wild birds, taken from the wild. They come in many colour mutations that are bred for and not found in the wild. Many are hand reared and imprinted on humans, and parrots can make wonderful pets for many people, being very intelligent and affectionate.
Your personal preference for pigeons is fine for you and I, but so is other peoples preference for parrots and parakeets. Scientifically, parrots are far superior in intelligence to pigeons, and for the OP a parrot or parakeet sounds a better fit for their needs, for example, they want to train their pet bird to 'step up', and use a perch like a parrot would. Sorry, but you can not really teach a pigeon tricks like 'stepping up', talking, using tools or playing with toys the same as a parrot.
A pigeon is NOT an ideal house pet. They produce a lot more poop than parrots / parakeets, they also produce a huge amount of feather dust and dander, which, apart from messing up your house, poses a serious threat to human health (pigeon fanciers lung), and for that reason they should never be kept in a bedroom for example. This is why most people keep their pigeons outdoors.
Pigeons are also much happier in pairs. A lone pigeon will get very lonely and frustrated when the human owner has to leave it, to go to work, shopping, etc. They form very close pair bonds and I would never advise anyone to keep a single one as a pet. I single one is also prone to develop behavioural problems out of frustration, like aggression, and trying to constantly mate with the humans in the house.
It's digressing from the point of this thread, but I can't help myself...
Every problem you state for a pigeon, is worse to a large degree for any equally large parrot. Cockatoos have much worse dander. Parrots are wild, exotic animals with a full set of wild instincts, social needs and behavioral requirements that indoor life struggles to fulfill, and should only be kept by the specialist who can provide the high degree of care they require - except for perhaps the budgie or cockatiel. They all make nice pets as babies, but the facts are most large parrots are rehomed within two years. There is a huge abandoned parrot problem in the USA. They are horrible pets for most people.
The pigeon is a domesticated animal and every species of parrot is a wild animal bred in captivity. They suffer in cages, are too smart for their own good, self-mutilate when stressed, scream, can bite hard enough to disfigure, become dangerously violent when hormonal, and also will try to mate with you, constantly, attacking you and anyone else out of sexual frustration.
Pigeons cannot hurt you if they try, they don't self mutilate, they do not scream. They have been kept for thousands of years and the fancy breeds are extremely calm and adapted to captivity. Their dander is manageable with vacuuming, dusting and air filtration and less than you get from a cockatoo. The droppings are small and should be fairly dry when the bird is well fed, have you seen what a macaw produces, or good lord, a lorikeet or other fruit-eating species in comparison??? Pigeons can be reasonably potty-trained. Mine uses a few select places, and does not soil her sleeping areas, our furniture, or our clothing.
They learn to step up and to recall by name very easily - it took my bird, a single indoor pet,
three days to learn her name. She comes when called from any room in the house and flies to my hand.
Large parrots are more intelligent, but this means they require extreme amounts of enrichment and activity or they self-destruct. Pigeons are still highly intelligent, can be trick-trained, learn their names, and LOVE to be touched, petted and coddled when they are raised with human attention. I have never known a parrot as affectionate as my pigeon, and I have met many parrots and owned several.
Bird-fanciers' lung is also seen in budgie breeders, and is ultimately an artifact of poor husbandry, insufficient cleaning, poor ventilation, or just keeping way too many animals. A pigeon loft is a wholly different environment than one or two pigeons in the house.
I will continue to recommend pigeons, and discourage parrot ownership. Even chickens, quails, and for the right person, even ducks are all better indoor pets than any large parrot - because they are domesticated. Living with a parrot is generally miserable, and it is not their fault, but they are just wild animals. Unless you can keep two, in a large space, and afford their expensive toys that they destroy, and expensive diets, and provide hours of attention, they should be left in the wild. I raised an orphan quaker parrot this summer and he is still a baby, but he is not a good pet. He bites, hard and all the time, he is destructive to anything he can reach, and he tries to mate with people's hands. We are looking to place him with someone who can provide a much larger home with other parrots - a typical house is no place for him to be happy.
But the pigeon thrives. We have become her social unit. She is never by herself, and has free reign. She is extremely affectionate but not demanding. I have kept 15 species of birds, but the pigeon is so far and above any other species in how well suited she is to be an indoor pet. It isn't just me. There are a lot of people with companion pigeons, and they will all echo my sentiment.
Now,
@Pogeonmomme How did you get your bird? She is beautiful.
This is my house pigeon Olive, she is the most rewarding pet I have ever kept.