Do you mean that your node is online at your home and has some custom software that operates as a server that merchants can send their payment requests to, and it requires authentication codes?
Ya basically. A protocol like this would need to be standardized of course, it's still not existent. But for privacy reasons, security reasons (phone gets hacked), and other reasons, I don't like to use a phone, and would prefer just to pay by paper, so if others feel that way, then we can create something, and it could be more accessible to a wider variety of people. No rush
Just brainstorming and making it clear because I keep seeing posts saying you need a mobile phone to pay by lightning, and you can't have anything equivalent to a paper wallet.
I don't think that there are any improvements in privacy here. As for security, a phone is actually way more secure than this theoretical software, because mobile OS are more resilient than desktops, and you are also turning your desktop into a server, as I understand. Also, vulnerabilities in your software could lead to your whole wallet being drained.
This system is really inconvenient for merchants, they would have to scan multiple codes, then wait for your node and hope that during this whole process there won't be any errors. It might take minutes, and in some cases merchants might not be able to afford to wait so long. It also doesn't make Bitcoin physical, because merchants need to have Internet connection, so users might as well just connect to their Wi-Fi.
Overall, I don't see such system being widely adopted, although maybe something like that will be used if both parties will need it for some reasons.