As an easier way to express this, there's a very simple statement to represent why cs is more important than kills (with a long ass explanation).
The most common misconception people have is that kills win games. However, if you look at this statement, there's something flawed about this. One, while kills at key moments are ultimately what allow you to take down turrets and the nexus uncontested or relatively uncontested, you have to first wonder HOW you get kills.
Kills are earned by 3 primary factors: good plays, bad plays, and gold difference. Good plays and bad plays will always fluctuate continuously with little control, but gold differences can be controlled and maintained, making them the only effective control for influencing kills. By getting kills, one increases the gold difference between themselves and the player killed. This makes further kills easier against said player.
This then allows you to simplify the statement that kills win games. Kills have to be earned someway, therefore you can conclude that the most stable way to get kills is to get more gold. Therefore, since gold gets kills, and kills win the game, then gold wins the game.
But then you have to ask, how do you get gold? Sure you get gold by waiting, but that's slow and inefficient, so doing just that won't work. You get gold by getting kills, but this is the hardest source of gold in the game. It involves taking out the most health, requires the most planning, and requires the most resources (health, mana, cd, time, etc.) than almost any other source of gold on the map, and it will only on average give 100-300 gold per kill. The average team will only get 5-10 kills within 20 minutes in total (on a decent match). This is only an additional 3000 gold at max, though it's more likely closer to 1.5-2.5k gold.
But then there are creeps. Probably the easiest source of gold in the game because they're just... plain... dumb. They walk in straight, predictable paths, do not retreat when low on health, and give 1/15th the gold a kill will normally give. This means that on average, every kill is worth around 15 cs. Now, the average team will earn 5-10 kills in the first 20 minutes, that's only 75-150 creeps. However, decent players will easily earn 150-250 creeps within the first 20 minutes by themselves. BY THEMSELVES. They have effectively earned the same amount of gold that the team earned collectively through kills, and then more, with NOBODY HELPING THEM. Multiply this by mid lane, top lane, and bot's ad carry, and you now have well over 4x the gold earned through kills earned through just 3 of the 5 players. The jungle probably has earned somewhere between half of that, and the support probably has no cs, so that's an additional 75-125 creeps approximately. That's still pretty much the gold earned by the team through kills at that point. And that's excluding the fact that the support probably has gp10's supplementing their income.
tl;dr: Creeps earn gold most efficiently, gold differences are the primary factor in obtaining kills, kills at the right moment will win games. Through the transitive property, one can conclude that killing creeps is the most likely way to win games.