Here's my take on the "problem". Are you applying a constant voltage? If so, you are victim of the speed-torque relationship. For raising, load torque decreases (fighting gravity hardest at the bottom) and it appears the angular speed increases near the top. Conversely, for lowering, load decreases (gravity assist is greatest at the bottom) and it appears the angular speed increases near the bottom. Some options are:
1. increase the motor gearing (electric screwdriver mechanism?) which makes the torque vs. speed relationship "stiffer" meaning the speed change less with load torque variations for a given motor voltage.
2. design some kind of mechanical counter-balancing system to present a constant load torque somewhat independent of bridge angle. then for a constant voltage, the bridge would rise/fall at a constant rate. you would apply a different voltage for raising and lowering.
3. install a motor speed controller. this is not just a so-called PWM motor controller though a true speed controller will undoubtedly have a PWM controller in it. all our speed-controlled train engines have this capability. a speed controller for a DC can motor does not need a tach for speed sensing - the DCC speed controllers in HO and G-gauge engines measure the internal motor parameters (so-called back-emf voltage) to measure speed.
options 1 and 2 will make it smoother but might have the problem of initially getting started for really slow speeds. electronics can kick-start the motor applying a boost to get things moving but implementing such a circuit might not be in your comfort zone. from the bouncing of the bridge when you stop, it seems you need to go slower than the video but as you say there is the kick-start problem. the advantage of option 3 is it takes care of all this - applying the necessary voltage independent of load torque including the kick-start to get it going.
the components for a DC can-motor speed controller are not expensive...maybe $10 for something that can power your bridge motor. the problem is i don't know if they sell one packaged in a configuration easily applied to your situation. that is, speed controllers are usually part of a system like an engine module that also controls lights, sound, smoke and communicates with a command-control system. i haven't looked on eBay - i'm always amazed at what you can find!
