I find this phenomenon to be especially prevalent in bot lane. A few missed cs can mean the difference between coming back to lane with a Pickaxe and boots or a BF Sword/Bligewater Cutlass. And depending on the ADCs in the game, that BF/Cutlass powerspike can determine the outcome of the rest of the lane. Trades won't go in your favor as often, meaning you'll have to play more defensively, which in turn can see you pushed under tower and prone to a 4-man dive. This will obviously lead to a down turret and perhaps even a dragon, which greatly snowballs the game in the enemy team's favor.
On the other hand, if one ADC has the money to squeeze in level 1 boots over another, the extra ms might mean the difference between escaping with your life or getting killed. And once the enemies have a kill lead AND a cs lead, it drastically reduces your ability to lane, forcing you even further behind.
But then again, the extra boots could allow you to dodge all-ins more easily from the likes of Thresh, Leona, or Blitz. Step out of the path of a key Blitz hook, and it immediately opens up the possibility of going aggro on the enemy bot lane, which could result in burned summoner spells or even a kill or two. And that could then translate into a free dragon or first turret gold for the team, which, statistically, provides a greater chance at winning the game.
Sure, missing 1 cs every so often might not be that influential, but the more mistakes you make in a game the greater the chances of you blowing any lead you manage to acquire. Most people don't fail to realize how important the micro-est of micro level play is to your success in this game. Every single decision, no matter how small and insignificant it may seem in the present, can lead to an uncontrollable landslide in the future; small mistakes giving way to larger ones, which inevitably spell defeat.
Personally, I wouldn't be so quick to dismiss the butterfly effect when it comes to an inherently chaotic system such as League of Legends. Ironically, you can never know with absolute certainty that those minor mistakes DIDN'T contribute to your loss. Best to air on the side of caution IMO.