You are obviously doing something wrong with that 15% win rate in the past 20 games. This post is probably a very large contributor to your poor win rate. Hitting the CS as a support does multiple things. I will make a list:
Messes up any kind of wave management your ADC tries to accomplish - If the ADC wants to put the wave into any particular spot to more safely farm, you make that impossible. You will always be pushed up under the enemy tower making ganks really easy for enemy and impossible for your jungler. This also makes any mistakes by either you or your ADC more punishable because you have a long way to get to your tower.
You take unnecessary damage - Any good enemy ADC will punish you like crazy when you are standing in the minions auto attacking them on Blitzcrank. You can't pull them because you are in the minions which will block it making you useless. You are melee and they are ranged. They can shoot you all day with no danger to themselves.
It is inefficient - As others have said, support items are cheaper and you get free gold for not taking CS as a support. You are there for utility on Blitzcrank, not damage. You can literally have 0 items and still do your job of pulling an enemy to you and knocking them up for the damage dealers on your team to kill while they are cc locked.
It tilts your team - Imagine if you were jungling and someone followed you around attempting to steal every camp from under your nose. You are basically doing the same thing to your ADC when you hit the minions.