So I've noticed generally that with some people (I haven't seen it on this site in particular though), when getting a tame bird, they would expect to be able to pet that bird.
But not every bird, tame or no, is a fan of being pet. This comes with the fact that not every bird mutually preens.
An example would be grass parakeets, such as the elegant parakeet.
They're very trusting in nature, but pairs do not mutually preen each other so don't expect them to want cuddles. At a young age, it is possible to teach them to enjoy being touched, but it's just not necessarily natural to them.
However, don't always expect birds that do mutually preen each other to always be a hands-on bird even when tame. While the popular Indian Ringneck Parakeet and related species are mutual preeners, they're known to be hands-off in captivity.
There are the special exceptions to that of course, a few being seen on YouTube (probably the reason behind their recent popularity). People may get these birds and expect them to be as cuddly as the ones seen on YouTube, but in reality they're not all like that.
Every bird is different. Even a species well known for their cuddliness such as cockatiels, may have a hands-off personality. Through all the years my mom hand-raised cockatiels, she would get a few babies that would bite upon any attempts to handle them.
All of this may seem to be obvious, so what is the point of this thread? Well, this is to say that it is possible to have good relationship with a hands-off bird. Even if they don't like to to be scratched behind the ears or under the chin (that is at least the case for my cockatiels), they can still be loving of their person in some ways.
When I first got Lucy, she wasn't so accepting of scritches, even after she learned to trust me. Of course, I'm sure the age might play a role in that; she was very hyper while young and could rarely hold still. But during the times where I we were getting to know each other, I've learned how sweet she could be even when she didn't want scritches. She learned to love hanging out on my shoulder or on top of my head (she wanted to be a barber at the time and always tried to style my hair), and enjoyed every activity I did with her, once she was no longer afraid of me (and I got her to step up for me). She always wanted to do what I was doing, and never liked it if I walked out of the room. I realized that this was her way of showing that she enjoys being with me. Over time she has learned to not only accept my scritches, but love them. Now there will be times where she'll ask for them when I ask her to step up.
Sometimes you may end up with a bird that will learn to accept scritches over time. However, that's not always the case, and learning to respect that is part of having a good hands-off relationship with a bird.
Of course it is important that the bird isn't afraid of your hands. Can't really teach a bird to step up onto your hand if they're afraid (and stepping up will make everything easier, especially if you have to get the bird off of something, or put them back in the cage). So if you wish to be on good terms with a bird that may not want to cuddle, work on teaching them that your hands aren't dangerous. You can do this by feeding them treats from your hand, and ultimately teaching them to step up. It will be a slow process, but it will make everything better in the long run.
We like to pet our animals because it is a common way we show our affection to them. Physical contact is a common form of affection in us humans, so it's natural that we may want to resort to it when it comes to loving our pets (of course like animals, there are people just don't enjoy being touched or hugged either), whether it's a dog, cat, bird or maybe even a turtle. And I have seen turtles that have learned to enjoy being pet too.
However it's not the only way to show affection to a hands-off bird.
Birds are social animals, and so they enjoy doing everything in groups. A way to strengthen a bond with your bird without petting them is to include them in some of your daily activities. Eating is a great example to this. Flocks will always eat together, so when you eat with your bird, it will simulate that of a bird eating with their flock members.
This doesn't always mean sharing food. There were times where when I'd eat, I would put a small bowl of pellets on the table so Lucy could eat with me. The act of eating while next to each other alone is enough to make it feel like eating together as a flock.
If by any chance your bird will sit on your shoulder, you can let them sit on your shoulder while you work on something. While some birds may prefer to try and "help" you with anything, some are content just sitting on your shoulder while you do it. It allows them to always be with you without having to try and fly (which is especially convenient for them if you clip their wings).
Disclaimer: Sometimes a bird may like sitting on a shoulder too much to want to get off. In that case you may want to get out of the habit, and simply let them sit on your hand. If you're going to sit down for a long time or at least be in the same general area, it would help to have a stand for them to sit on, or just somewhere nearby (just make sure there's nothing the bird shouldn't chew on that they most likely will).
The best thing you can do overall is simply respect the bird's boundaries, as pushing them to try and enjoy petting can make things worse.
Some people will train birds to get used to being picked up, held and touched. While this may be useful in a case where you may need to move the bird during a time they might not to step up, it's also good to have a relationship with your bird where they don't feel forced to do anything. If you decide to try and condition your bird to being touched or held, I would advise doing it gradually, and with positive reinforcement (clicker training is effective for this).
This is all the advice I can think of for now. I hope it's helpful for someone. Also if you have any advice of your own on working with hands-off birds, do feel free to share.
Just thought it'd be nice for people to know that there are ways to bond with a bird even if they don't allow you to pet them.
A lot of the things mentioned are ways people bond with hands-on birds too, but I feel that it's a good reminder that loving your bird doesn't always mean petting them.
It's funny that Shodu and Buster seem to be polar opposites of each other with certain behaviors, and yet Lucy may either meet perfectly in the middle, or act like neither of them. I can't quite tell who she takes after more.
