Does anyone sell a direct-drive recumbent with no handlebars?
To the best of my knowledge there are no suitable direct-drive hubs available commercially, which means that there are no bikes sold that use them. Schlumpf make a unicycle direct drive hub that offers 1:1 and 1:1.55 gears, but for a recumbent bike those ratios would be ridiculously low. Also, it's obviously fixed gear so without a freewheel it would be even less useful on a bike.

You may be able to convert a standard hub gear to direct drive, using a larger axle and feeding the drive through the centre. Hubs are designed to have drive from one side, and pedalling torque is quote large so this would probably involve a major redesign of the hub (I suspect this is why so few people have built them). Another problem is that hub gears are designed to have 2:1 or more gearing up from the chain drive before the hub. So once you got it working, you'd have lower gears than you expect. Using short cranks is a partial solution to that at best, but it's the only thing I can think of that might help. Greenspeed suggest that 100mm cranks are quite usable with only a slight loss of power (but higher rpm = lower gearing needed)



Your second question, about having no handlebars, is much easier to do. Most front wheel drive, moving bottom bracket bikes can be ridden hands-off, and it's relatively easy to learn to ride without the handlebars at all. The problem is that most countries require bikes to have handlebars (it's an offence to ride without holding them).

One problem I see with the design you link to is that it's partially rear wheel steered. That will affect stability (making the design somewhat harder, but if you're buying one they've already solved that problem). But it will also affect turning. To turn away from something, the rear wheel must move towards it. All rear wheel steer vehicles suffer from this problem. One partial solution is to use a tadpole tricycle configuration, so that the rear wheel is between the front wheels and thus can move sideways as far as one front wheel when turning away from a kerb or other barrier. For a bicycle, the solution is to stop and lift the bike away from the barrier. The problem is that if you run up next to a barrier at speed it may be difficult to stop without hitting the barrier. That's annoying if it's a kerb and you end up sliding along the footpath, but can be dangerous if it's a wall or hedge.

With unusual bikes that someone is selling, I find it's helpful to ask how much someone other than the inventor has ridden one. If they've sold a few and someone has ridden a couple of thousand kilometres on one, then great, it's probably usable. But if it's a great project by one person and you're going to be the first real customer... be very careful.