I almost never approach Merlin and attempt to touch him with my fingers. I want to be able to hold my hand right next to him without him feeling the need to scoot away, and I practice this as often as step up or anything else (I wasn't able to do this with the little owls I worked with, and I wish I had, because ONLY approaching them to put them in a carrier taught them to hate me). He
has started enjoying scritches in an odd sort of way-- he will fluff up and push his head against his favorite toy in the posture a budgie would adopt while another budgie is preening them. At first, I moved the toy a little to see if he liked it. Then I just started scratching his cheek myself! He goes back and forth between play and preening mode with his toy, and does the same to my fingers-- beak boxing and growling and preening. A few times he peeped when I stopped, as if to say "more!" and a few times I've misjudged and he gave me an (irritated? playful?) open-beaked little bump.
He seems to really enjoy when I stand close to his cage (reminds me of my extroverted little brother who hates doing anything alone-- "want to swim?" "no" "okay will you come watch me swim?")-- sings and becomes more playful-- and has recently started interacting with my hands more (best was a night ago when he stayed on my fingers and was singing!

I could feel all the notes and chortles buzzing out of his little body!) but he doesn't care to be on my shoulder or step up when he doesn't see a treat. For some reason, using the toy as an intermediary just made this odd connection in his brain; until that moment, I don't think he understood my few gentle touches as attempts to preen. And he still doesn't posture this way for me-- oooonly the toy.
I've read older threads on here where folks say some budgies accepts this kind of petting when they are juveniles, but are even more hands-off as adults, too. They can also have individual preferences about where on the head and cheek they are okay being touched.