One point has been missed in the other fine answers.
On an aircraft carrier, the nose gear of each aircraft has a pull bar installed that links with the catapult sled. This pull bar is manually engaged by a deck crew member for each aircraft that is to launch.
In order to do this at an airport, there would need to be a ground handler stationed at each runway turn-off point to engage the pull bar to get the aircraft moving. I pity the poor souls who would have to stand next to a runway at KBOS or KJFK in the wonderful weather they are expecting this weekend.
It's possible that this system could be automated, but that's another very expensive design element that has to be brought into play, tested, and receive CAA certification.