DS either drops straight to sleep mid-feed or mid-whatever he is doing, or he screams and screams until we pick him up. There’s no mid-point - if he dropped off after 5 mins we’d leave him to it, but he just gets more and more worked up and hysterical and holds his arms out to us to be picked up, so it would feel cruel to just sit and watch him IYSWIM? Patting and shushing makes him worse not better, he tries to grab onto our hands and pulls us towards him. Doesn’t sound like yours does that, or like you would leave him crying if he did.
I’m not really a goo-goo sort of person either, but things I’ve really found nice to do during the day with DS are baby swimming (can start any time from birth but realistically they’re more likely to get something out of it at 3-4 months), “dancing” with DS (Beyoncé on YouTube, waving his arms about in time with the music, singing along with the chorus), and baby music classes (I go to one called Hartbeeps, it’s a franchise so might be one near you).
Also wearing a sling - I bought it because my local station has loads of steps and no lift, and I couldn’t get the buggy up and down easily on my own. But it’s really comfy, and I’ve taken him out on long walks, round museums and all kinds of places in it. He can sit up and look around, I can chat to him and don’t have to drag a buggy round with me. I also noticed I tend to ignore him in the buggy because it’s front-facing and I can’t really see him.
Feel free to ignore, but I’ve really enjoyed those things because I’m close to DS while I’m doing them and we can have fun together, but they’re structured activities so I’m not sitting looking at him wondering what he wants me to do next.
Newborns don’t do much, but once they're past about four months they do start to play with toys and interact with the world. You’ll probably enjoy that phase a bit more.