BF Sword NEEDS a component item
TL;DR The items to build the Infinity Edge have less risk behind them with varying prices and power then the Bloodthirsters low power, low cost and high power, high cost component parts. If the BF Sword has an item it is built from then the Infinity Edge will not be the dominant AD first buy item all the time. (Make it the pickaxe because it would fit)
Most of us know that the Infinity Edge was recently nerfed because of its power and popularity. The Bloodthirster is supposed to be a good counter to it with a consistent healing effect to negate the critical damage, along with a bubble shield and healing off minions/jungle creeps but building the BT creates a risky situation before it's even built. (Essence Reaver/Mercurial also fall into a similar issue but BT is picked more so the BT will be explained. Two birds with one stone in a way. Maybe three)
The Infinity Edge comes in three parts. 730 Cloak of Agility, 875 Pickaxe and 1550 BF Sword. When laning turns bad and you are forced to recall or you die, the Pickaxe is a common first buy. It's decent damage, boosts the damage of AD scaling abilities and helps with last hitting creeps. If laning works in your favor then you will probably get the BF Sword first for more damage. The only time you would not be able to do this is if you die early, can't kill champions or last hit properly.
The Bloodthirster comes in two parts. 800 Vampiric Scepter and 1550 BF Sword. Following in the IE's tracks it seems simple. If you can't get the BF Sword when you are back at the fountain then get the Vampiric Scepter. The problem with The Bloodthirster starts with this. 10 damage and 10% life steal seem useful for sustain compared to a pickaxe but that 10% is reduced with a champion/minions armor and also heals based on your own damage. A half way leveled ADC at level 9 has an average 75 base damage. Add in 10 damage from the Vampiric and you get 85. Reduce that 85 from the targets armor and then 10% of whatever is left over is what you're healed for. Plus you will probably not have 75 base damage because you will not have reached level 9 by the time you are forced to buy the Vampiric Scepter in the first place.
But this means the Pickaxe is in the same boat isn't it? The pickaxe has 15 more damage over the Vampiric Scepter which is also reduced as well but it's a difference of 15 (ad) and 10 (ls). Reduce them both by the same percent and the 15 is still going to negate 10% when the enemies total AD is not even at 100 yet.
** What would happen to the Infinity Edge if the BF Sword was built out of something?** If it was built out of a Pickaxe then it would allow the Mercurial, Bloodthirster and Essence Reaver to be more desirable items to be built early on. You would not have to sit and wait for 1550 gold when you need to have items to win your lane, if not defend it. The IE would just boil down to someone not thinking and 'wasting' 200 gold on a pair of pickaxes rather then upgrading the first pickaxe into a BF Sword. Not that you would lose anything other then an item slot and gaining the option to change one of the two Pickaxe's into something else, if the situation calls for a different item, over having a BF Sword that is just waiting to be turned into the Infinity Edge and not having enough gold to instead, purchase a needed item. (Frozen mallet, Zephyr, Maw of Malmortius, Last Whisper)
So as another sort of TL;DR, Make the BF Sword build out of a Pickaxe and the power of some item parts over others will shift significantly. Powerful items will not be ignored due to a players lack of funds from poor laning and we shouldn't see another item nerf because it got too popular from the power of its parts alone.
how it builds from
and make a
or maybe two build into 
gives you lifesteal with comparatively low investment.
, but AD has building items in