I think doing zero kills flat out is a poor idea. Lets do some math.
You want to get a Philosopher's Stone as your first item, a rather useful Gold Per 10 item. This is 800 gold.
Now, lets assume you have greed also, this will give you 15 gold per 10 seconds. If you do nothing else then rush a Philosopher's Stone, use all 475 gold into getting it (A somewhat poor idea), it will take 217 seconds after minions spawn to gain enough gold to get it, or 5 minutes and 7 seconds after game start.
But, getting nothing but the items that build into a gold per 10 doesn't tend to work out. So, lets start with a more likely starting build. A Faerie Charm, some Wards and Pots, lets just assume you spend all 475 gold. Now, with only a Faerie Charm, it will take anther 620 gold. To get that much, it will take 413 seconds after minions spawn to get enough gold, or till the 8 minute 23 second mark. Now, this is assuming no kills, but lets assume you are having a poor game where you are dying quiet a bit or just can't do anything. Having nothing but a Philosopher's Stone roughly 8 minutes into the game can make it so you won't be able to help as much.
Now, the easiest solution is to kill some, but not many minions at the start, get the stone, and then let the AD carry farm. The rates at the start of the game is 16 for caster, 22 for Melee (increases 1 gold per every 5 minutes after spawn)
Lets average the two to 19 per minion. So, every minion kill will shave off 12 2/3 seconds to get a Philosopher's Stone..
With needing all 800 gold, it would take till 10 minutes 23 seconds to get 800 gold with no minion kills. Needing 620 (Start with a Faerie Charm), it will take 8 minutes 23 second into the game to get enough gold. Needing 365 gold (starting with Regrowth Pendant), it would still take 5 minutes and 33 seconds to get enough gold.
Now, if you started with a Faerie Charm, and killed 1 minion per wave (Which spawns every 30 seconds), you would be able to get enough gold at roughly the 6 to 7 minute mark. If you were able to get enough gold at the 6 minute mark, you would not only have the 190 gold for killing the minions extra, you would be able to get anther 70 gold in the time between when you would normally get the stone and the accelerated time.
Now, you can argue that the 190 gold would be better for an early game AD carry, but allowing the Support an extra 260 gold can be put into several wards, which can keep people safe and better allow for kills. Or they can be invested into a second gold per 10 item, like a Kage's Lucky Pick, which will net you anther 65 gold in the saved time.
tl;dr
Allowing for some early minion kills can allow a support to to accelerate his build, making him more of a use to the team mid game, at a "minimal" loss to the AD carry. This allow a more safe mid game in the event the AD carry gets shut down early game IMO.