As a current bronze player *sad face* , I have observed that other bronze players are not lacking in mechanical skill, not by a long shot. What holds bronze players back is the inability to think of the macro game, leading to a severe case of team disjointedness.
Yes, kills are important to get, but they should always be a consequence of executing a strategy with your teammates in mind, particularly outside the laning phase. Racking up kills doesn't mean squat if you can't reach the enemy nexus at game's end.
Mistakes in this game are punished too heavily for players to do their own thing with no awareness whatsoever. Chasing without vision, bullying lane and then being surprised when you are ganked, no concern that all enemies are off the minimap; all this is near impossible to overcome by out-mechanicalizing the enemy.
Giving just a little thought as to what the enemy could be planning/doing as well as a little common sense (not overstaying so that the enemy has a numbers advantage for example) goes a long way, a very long way. League is a game of strategy. Mechanics merely allow you to execute said strategies.