I fantasize about it too, from time to time. Maybe it is the nostalgia of having single speeds when I was a kid and those were the years I had the most fun and feeling of freedom on bikes.
I also remember when I got my first multi-speed bike and how much easier that made the riding.
My last single speed was a BMX, so I would stand up and mash to climb hills and coast down them.
Unlike you, I think I would give up the freewheel sound for a drum rear coaster brake. I never managed to wear one out. That way, the only cable left on the bike would be the short one for the front brake.
Yes, big hills would take the fun out of a single speed, for me, unless it was geared quite low or I was willing to walk up them.
I have one single speed eBike. It's geared for 15 mph and the electric aspect takes the pain out of hill climbing in too high of a gear. But it adds back in the complexity you're maybe looking to avoid.
Have you seen the single speed bikes the high school kids are riding these days? They look like very large BMXs. They're geared low so aren't fast at all, but their thing is that they're learning to ride wheelies everywhere on them! Fun, I would have been SO into that.
I think if I were to get a single speed bike nowadays (other than the eBike) it would probably be a build based on
the Surly Steamroller frame. I'd probably do a drop bar and gear it for maybe 12 mph. Coaster brake in the rear and a simple rim brake in the front. Belt drive, to really keep it clean. Or alternately, you could keep the (outward) simplicity and fit a 3 speed internal geared hub. You could still have the belt drive, too.