When is it better to get gold per 10 items instead of saving for something else?

I know supports usually get Philosopher's Stone and Heart of Gold, but what about other roles? Also would getting a 3rd gold per 10 item on the support such as Kage's Lucky Pick be a good idea or would it be better to get one of the other support items?

The point of gold per 10 items is to increase that gold generation early since you will not be getting creep score. You get a philosopher for mana regeneration and heart of gold for health bonus for your survivability. Generally you want to build a gold per ten on a character if it is beneficial for you in what it builds into generally. Stone builds into Shurelya's Reverie, heart finally into something useful, Locket of the Iron Solari. These are both useful to a team as a support. Kage's pick only builds into deathfire, not as benificial to team, and Avarice Blade also gives gold per ten but builds into ghostblade which only gives benifits to holder again. You would want to sell pick and avarice blade later on for other items due to this fact which is not very viable. From the league wikia, the items pay for themselves after roughly 12 minutes. The gold you have to put in is locked up for 12 minutes each so there are more viable options to buy.

I like to build 3 GP10 items on my supports because when coupled with all the gold generating runes and masteries, you get 3.525 gold per second. That's a Sight Ward every 21 seconds or an Oracles Elixir after 1:54 seconds.

Being a warding machine by about 20 minutes into the game is so valuable that I would rather hold onto these 3 GP10 items for as long as possible and not worry about what they upgrade to.