I don't mean to seem rude when I say this, especially since I am only making an inference based off of what you've written here, but I honestly don't think you should be focusing so much on "which species of bird is more cuddly." For two reasons:
1) I have two male cockatiels. One is a little older than the other. One of them couldn't care less about head scratches. He hates being touched/handled in general. My other cockatiel LOVES getting head scratches. When he wants head scratches, he'll lean forward on his playstand and stare at me with puppy-dog eyes until I ask him to step up. He then puts his head down and waits for me to start petting him. Even so, he is NOT a cuddle bug ALL OF THE TIME. Today, for example, he's been a little fussy. It's probably hormones. Or maybe he didn't get enough sleep last night. But the point is, even when he is being moody, I can still be with him and have him out of the cage and STILL not get bitten. How? I pay attention to his body language. I respect him and his space on the days when he wants me to leave him alone. And that's fine. The point I'm trying to say is that, even if you have a bird of a particular species/gender, every individual bird is different. But, even more so, you could have the SAME bird act VERY differently over the course of ONE day!
2) It sounds like you seem to struggle with reading bird body language. Birds often warn you with their body language if you are doing something they don't appreciate. Depending on the species and the individual bird, it could be obvious or it could be subtle, but that is why you need to pay attention and watch for the signs. However, if you ignore those signs and continue doing the thing that the bird doesn't like (such as petting), the bird learns that you only get the message when they resort to biting. You can, in fact, teach a bird to bite.
It sounds like that's what you've done with your phirrhura (I'm assuming you mean a small conure, right?) and your linnie. And whether you get a pionus or a senegal parrot, the same thing is going to happen with your new bird - you'll accidentally teach the bird to bite.
Like I said before, this is an inference based off of only what you've posted here. And there may be a very good chance that I've misinterpreted you, and if I have, I apologize. But I truly think that if have a biting problem with your phirrhura and your linnie, getting a new bird isn't going to fix it. I really think you should focus on rebuilding the relationship you have with the birds you have right now. Then maybe consider getting another bird later on.
I hope this helps. We are all here to help you if you have any other questions.