I'm probably speaking out of turn here as I am a cat person not a dog person, but why does a puppy need a stair lift? None of my dog loving friends has ever expressed any concern about their dogs being injured going up or down stairs.
the simple answer is that because they have a dog and not a cat !
I too agree that the sled, the rails, the motor and pulleys and such do not need an Arduino.
what you can do is to have a switch, or series of switches, that sense the weight. 14 pounds work, 30 pounds does not.
I would even move the switches to make a front ramp come up, even a small one. like a D-Day landing craft.
it would train your puppy to sit and wait and not jump out.
and at the top, it could cover any of the extra parts.
I would use a pipe, like electrical conduit as one of the rails so you can have a way to stop the cart from coming away from the rails. you would not need rollers, just some plastic like a dollar store cutting board.
you are also correct that you do not need a massive motor. you can get a gear motor for under $20 that would work. you trade speed for power. one of those 2,000 pound winches typically might have a power window motor in it, with a lot of gearing.