Yes, you can get all the cheap cup and cone fixed/single-speed hubs and wheels you want. The hubs are still mass produced (unless they made so many during the fixed gear craze that they're still coasting off that, but I digress) and used as OEM on bikes and aftermarket wheels. Examples would be the Formula TH31 (fix/free) and TH32 (fix/fix) and Joytech A166DSE. For some reason KT is no longer listing your cup and cone Quando hubs on their catalog or site. I don't know what the story is there. In the US, one of the major distributors and wheel producers, J&B Importers, makes a bunch of wheels with the aforementioned Formula hubs, and any shop can get them for you.
The track hubs from Shimano and Campy are also cup and cone, and very high-end. Because of their lack of sealing, I've never loved the idea of getting them just to ride on the road unless you live somewhere without weather or dirt. But they are some of the most exquisitely made cup and cone hubs around, if you're into that sort of thing.
You really don't need anything to do a fine job of replacing the bearings on the common cartridge bearing track/singlespeed hubs beyond a hammer, a rag, some sockets and washers, and a block of wood. That sounds crude, but the fact that they use a solid, shouldered axle design means that there's no way of getting a special press on them anyway, and no bushings that are going to do anything for you that the old bearing can't.