I will use the 757 as an example, but everything is true for every multi-axle landing gear I've ever been around (except the 777).
The truck (bogie) is tilted into a specific angle so that it'll fit into the wheel well, all other functions are secondary. The truck must be angled because the landing gear strut itself is not completely perpendicular to the fuselage, due to other considerations. This is what the 757 Maintenance Manual says about the truck position actuator (aka tilt actuator): "This actuator applies the force to put the truck assembly at an angle to permit clearance with the structure."
This YouTube video shows the extension/retraction of the 757 landing gear, you can clearly see in the video that if the truck was not at a specific angle it would impact either the keel beam or the wing to body fairings.
The reason it stays tilted in the down position comes down to cost and complexity. The actuator is a simple one-way hydraulic actuator, meaning it can only be powered into the extended direction. Any time hydraulic power is supplied to the landing gear, either in the up or down position the truck positioning actuator is powered on the extended side, regardless of gear position.
Extra information, for the non-TL:DR crowd
When the aircraft touches down, the actuator attempts to keep the truck tilted, which massively increases the pressure in the actuator (normal system pressure is 3000psi). A check valve prevents the hydraulic fluid from being forced back into the hydraulic supply system, then at 4500 psi dumps the fluid into the return allowing the actuator to collapse. At takeoff when the aircraft rotates it again forces this actuator even further into the retracted position as the aft wheels get closer to the fuselage. As you can see the stroke of this actuator must be quite long. So fully extended gives you a toe up position and fully retracted gives you a toe down position.
To have the truck powered to any position other than fully extended would first require the ability to power in both directions, which adds a lot of complexity to the hydraulics system and actuator and this would only get you fully extended and fully retracted. To position the truck in any other position would require complex control valves with position feedback or multi-position actuators and also sequence valves to ensure any time the gear was not down and locked the truck was in the proper position. Since there is no need to flatten the truck for landing, this is not down.
Finally having the truck position always powered fully extended means the truck will always be in the correct position for retraction and you can use the truck tilt to detect air/ground (weight on wheels). In fact on the 757, if pressure to the truck positioning actuator fails, the aircraft will not switch to ground mode until the nose gear shows weight on wheels.
To address the nose down moment that has been mentioned, while technically this does happen the moment is very small relative to the aircraft and is over in a fraction of a second. The actuator itself is fairly small, smaller than a spoiler actuator and reacts its load on the strut very near the truck pivot pin, so the moment arm is short. Also this same force would fight against aircraft rotation at takeoff.
The 767 does nose down because its struts are at a different angle than the 757 and they have to be nose down to fit in the wheel well, this is a design decision due to a limitation at LGA.