League-of-legends – Which of the 3 support items can earn you the most gold

When to buy those items is very situational:
Assuming generic AD carrys like Graves, Sivir, Miss Fortune or Ashe, the first time you go back to base you should buy 1-2 Dorans Blades and a few Healing Potions instead. Those items help you stay in lane longer with their health, lifesteal and damage bonus, which allows you to farm more gold before going back the next time. They are what empowers you to actually get that B.F. Sword.
Don't buy Berzerker Greaves first because they don't actually help at all. You won't be attacking constantly but in bursts instead, so the Dorans Blade that you should buy instead actually has a bigger effect on trades.

If you can stay in lane long enough because the opponent doesn't harass or is completely outclassed, you can get the B.F. Sword immediately, but that rarely happens in a normal game. You need the small edge that Dorans Blades give. If you come back to lane without a Dorans Blade and your opponent has two of them, he can completely bully you around and free farm on a frozen lane while you sit under tower, not getting any farm or experience.

Also, if you can't afford the biggest component, for example for the Infinity Edge, you can get a smaller component like the Pickaxe first (after Dorans Blades), which still builds into the same item and makes you stronger in lane than if you would wait for the big item.

"Selling items is bad" is mostly true, but:

• If the item you are buying now is helping you to stay in lane longer (i.e. Dorans Blade/Dorans Ring) then it is worth it because by staying in lane longer, you gain more gold than you lose from selling it.
• If you come back to base lacking 200 Gold for your next major item (Infinity Edge, Bloodthirster, Phantom Dancer), going back into the fight without it would make you weaker now and potentially put you further behind than selling one of the Dorans Blades and getting the big item now.

It's all about short term benefits giving you an edge to have a bigger advantage later.

All of this is true in the same way for AP carrys with Dorans Rings.

You list primarily tanky supports, who were hit the hardest with the removal of Heart of Gold. This leaves two other GP/10 items: Philosopher's Stone and Kage's Lucky Pick. I ignored Avarice Blade as I don't feel it compliments the support role.

A Philo stone can build into Shurelya's Reverie, Eleisa's Miracle, or Ohmwrecker. Shurelya's is still a great support item, and Eleisa's reduces the cooldowns of the summoner spells Heal, Clarity, and Clairvoyance. It can also free an inventory spot if purchased early enough! Ohmwrecker is outshined by both items, and I don't recommend buying it on a support.

Kage's Lucky Pick turns into Shard of True Ice, Twin Shadows, Morellonomicon, or Will of the Ancients. The first two are most viable on a support, and the Shard continues to provide GP/10.

So a lot of things affect whether GP/10 is still viable.

For a Philo Stone:

• Did you take Heal or Clairvoyance?
• Do you need extra sustain?
• Does your team need a Shurelya's? Is anyone else buying one?

For Kage's Lucky Pick:

• Does your champion benefit from extra AP?
• Do you need additional CC?

Another new item is the Sightstone. For an investment of 950 gold, you can place up to 2 wards at a time, with each ward lasting 3 minutes. The Sightstone holds 4 wards total and refills when you return to base. This breaks down to 50 gold/minute, and starts saving money after 19 minutes (assuming you immediately place wards when available). It also provides +180 health, making it a strong choice for tanky supports.

Previously, GP/10 items were essential to compensate for the cost of wards. This is largely mitigated with the usage of a Sightstone, freeing money for other items.

A Philosopher's Stone costs 700 gold and pays for itself in 23.33 minutes. Kage's Lucky Pick costs 765 and pays for itself in 25.5 minutes. These times are halved if you sell the item. Because it takes so long to make money from GP/10 items, it's best to buy them as early as possible.

Compare this to the gold granted from kills/assists. A champion who hasn't been killed is worth 300 gold, and 50% of that for an assist. The more times a champion is killed, the less money the kill rewards, with a minimum of 15 gold.

Since 2 GP/10 items provide 60 gold/minute, you have to get a 150 gold assist every 2.5 minutes for assists to equal GP/10 items. You can get an additional 5 GP/10 from Quints and masteries, resulting in 90 gold/minute and needing an assist every 1.66 minutes.

The new pick-pocket mastery offers an additional avenue to generate gold. Unfortunately, this isn't very viable on melee supports.

While supports are no longer required to get GP/10 items to support expensive ward-buying habits, these GP/10 items are still cost-efficient for longer games AND they upgrade into items that are very useful for most supports.

Ultimately, the decision to buy GP/10 items comes down to play-style and preferences. If you prefer to play aggressively and need extra stats to stay alive, skip GP items. If you're trying to maximize gold gained, GP/10 is still the better choice (unless you can get an assist every 1.6 minutes!).

tl;dr: GP/10 is still very viable, especially since those items build into new support-focused items. However, it's no longer a necessity since the Sightstone makes warding much cheaper.