I disagree with some aspects of this.
The further you push, the more money you get as the greater the multiplier value. If you're pushing, it probably means you're winning. If you're winning already, why would you have a mechanism that lets you snowball even harder? How is the defending team able to recover from such deficit?
Secondly, the point of getting kills is that you prevent them from farming, also denying them opportunities to defend objectives such as dragons, barons, towers, buffs, etc. Killing someone doesn't mean just gaining 300 gold, it allows for snowballing opportunities and objective control.
For example, as a jungler, I want to gank bot lane or the enemy mid player. If successful, it usually leads in taking the dragon. That is how you snowball. In regards to last hitting, sure it can be tedious, but there is more to it than just getting gold. A lot of poke and zoning is involved during the laning/farming phase. There is skill required and the more skillful the player, the more advantage they will come out with after the laning phase. If you trade well, you can send the player back to heal at fountain, and you can free farm while they are traveling back to lane.
Your concept is a great idea but I find it lacks depth.