I've bolded my comments that answer your assertions, leaving one. "It is a piece of paper...."
No, a "paper wallet" is not a piece of paper. It might be, but it might also be engraved phrases on steel, or other materials. And there isn't necessarily just one copy in one location.
One more time. A paper wallet is simply writing down and storing private keys.
Sure. Paper wallets are basically private keys that are stored offline.
A paper wallet isn't just simply writing down and storing private keys. The paper wallets would have to be generated and spent somehow. If it somehow gets into contact with a computer that has at least been connected to the internet, it isn't considered air-gapped. It can be secure, if you generate and spend it securely. But the whole process is basically spinning up an offline desktop wallet which I would consider just using it instead of going through the whole fiasco of having to store it and spend it securely.
If you do get any malware on your desktop, your private keys could get compromised, desktop wallets or not. I don't find it particularly reasonable to say that a paper wallet is more secure as a desktop wallet when it isn't generated securely most of the time, regardless of how big of a risk it is.
It's not wrong to say that paper wallets are secure. But the thing is, most people generate it and spend it online which makes it much less secure that it seems.
I'm having some trouble parsing your comments and trying to understand in what possible context they might be true.
I think you are saying that sloppy use of paper wallets is no better than an online wallet.
However, I can't agree with that. Suppose you have a sloppy and careless user, who may have malware. He creates some paper wallets and five years later remembers he has them, and does something with them.
Surely he is better off than the sloppy and careless user who had bitcoin on his malware infested computer for five years.