As anyone can easily conclude, having a driving front wheel on a traditional bicycle would be mechanically more complex than achieving the same result with the rear wheel. The front wheel is primarily used for steering, that is it rotates around a vertical (almost) axle. Relegating the steering to somewhere else (i.e. to the back wheel) is not an option as it makes a very unstable system (riding a bicycle rearwards proves it). The rear wheel is either fixed relative to the cranks, or at least stays in the same plane as the frame.
Side note: children's tree-wheelers and ancient penny-farthings have pedals directly attached to the front wheel and do not suffer from this problem.
So, any design that has a driving front wheel needs extra complexity to overcome this issue: to deliver power from the pedals in the middle of the frame to the wheel. This is doable, but results in more components, more weight, less reliability and less efficiency. If we only want to power one of two wheels, the rear wheel wins.
Now, suppose we want to have a design where both wheels are powered by the rider. It has to be noted that the golden rule of mechanics mandates that no advantage in power will be achieved: the same power generated by the rider will be distributed over two places, instead of only one (plus losses on friction, higher in the case of two drivetrains). If there is any potential benefit in having two-wheeled drive, it has to lie somewhere else, e.g. in increased control over the ride.
Side note: we do not consider additional "pedaling with hands" here. It would theoretically give the rider more power at the cost of less steering control.
Let's consider two categories of bicycles: road (primarily racing) and off-road (primarily MTB but also e.g. fat bikes in snow)
road bicycles are ridden over pavement of different quality. Both tires touch the road all the time. The traction between tires and the surface is already good enough to transfer all power generated by the rider into forward acceleration of the bicycle. The second drivetrain won't change or improve over that.
off-road bicycles are ridden over uneven or loose terrain. There are often moments when only one wheel touches the ground, while another one is briefly in the air or has otherwise lost traction. In such conditions, having the remaining wheel as driving does become useful. In conventional designs, rear suspension allows to increase traction of the driving wheel by allowing it to better follow the terrain's profile. Simply speaking, when encountering a bump, the wheel, instead of jumping off and flying into the air, stays closer to the ground and regains traction faster.
It would seem to make sense to have a full-suspended and dual-wheel-driven bicycle to keep both types of benefits off-road.
This is becoming really involved at this point. Modern rear wheel suspension designs are extremely complex and finely tuned, partially because they have to deal with the chain that limits their efficiency (see pedal kickback). Repeating the same thing at the front wheel is possible, I would assume (e.g. with a linkage fork instead of a traditional telescoping one), but again it results in additional weight and complexity.
Who knows, maybe one day such designs will become available to athletes and regular consumers, once (and if) other options to improve bicycle capabilities have been exhausted. In the end, the bicycle design is limited by human abilities: there is no point in making a bike with crazy high traction if no human with enough power to climb it exist.