Heh - recumbent stationary bikes are modeled after the real thing. Riding them is different, but they're still bikes. Just like with upright bikes, speeds vary greatly; and you can get ones that are slow or ones that are amazingly fast. The UCI didn't ban them from competitions because they were slow or because they looked funny, the did it because they were deemed an 'unfair advantage!'
Recumbents tend to be more expensive than uprights, but you should be able to set your sights relatively high if you're willing to look around. For dog-walking, I would be leaning toward a trike; but it might be harder finding a trike that is both speedy and inexpensive. An older long wheelbase, like a TourEasy might also be a good choice. A leash could be attached to a rear rack, keeping it well away from your pedals; and the long wheelbase would make it more stable.