What does "the dog who caught the car" mean?
Solution 1:
Maybe some people use it to mean "a goal was attained and now they don't know what to do", but in this context Harris isn't saying that. He's saying that a dubious or worthless goal was attained, regardless of what happens next.
If you race your friend to a chain-link fence and you win, only to find out the fence was electrified, you're the dog who caught the car.
For the dog analogy, either it dies or nobody in the car is concerned. The dog's aggressive attempt to achieve "catching" the car is a miscalculation. There is zero intent to convey a goal or achievement was reached, because the goal was to attack the car which is the silliness implied in the statement. The "dog" is looking for a fight it doesn't know it cannot win.
You can infer a "what now?" effect, but the gist is, it's playing out of its league and doesn't know it.
Solution 2:
the dog who caught the car wordspy
n. A person who has reached their goal but doesn’t know what to do next.
My sense is that the dog is not dead ... he has achieved a long desired goal ... now what. Colbert's quote is apropos for a humans: having caught the car (the office), metaphorically they now have to drive it.
Another reference:
▶ the dog has caught the car. The New Partridge Dictionary of Slang and Unconventional English, 2nd Edition
a person (or group of people) who has achieved a goal and is now at a loss for what to do next. AmE
As in:
It will be very difficult for Central Command to calibrate its war plan to everything taking place in the country now. The dog has caught the car. — Retired Major General Don Shepperd, CNN, 11 April 2003
And a third reference: The Complete Idiot's Guide to Weird Word Origins