I almost always build BT first on ADCS and here is why.

I am going to start of by saying that I am not a good player, but i think my logic is good at least for my play style. TL;DR AT BOTTOM Also i think i am missing a ton of points, i just can't think of anything else right now.

Pros of IE first

• crit chance is king for ADCS
• has the highest possible damage output from a BF built item.

Pros of BT first

• self sustaining
• Sustain is king in lane
• Cheap BF build
• ability to stay on the map longer ( more farm more cs)
• more forgiving

BT vs IE the argument

1. Price argument

• BT is 300 gold cheaper than IE.
• You get your second item (PD/Shiv) faster ( you also hit your 2 item power spike earlier.

2. The Laning phase argument

• Sustain is king in lane, therefore BT>IE (you can trade, maybe lose cuz of crit, then farm 2 waves of minions and get full health again or kill golems or gromp) [PLUS gromp damage nerf makes it so you don't lose health when attacking it with BT]
• IE only only adds 100 additional damage, statistically every 5 shots.
• BT heals 100 damage every 5 shots, but additionally passive gives you 160 hp shield at level 9. scales up to 350 as you level.

3. The Mid Game Argument

Pros

• BT reaches 2 item power spike much faster.
• Gives you the option of going IE or LW 3rd item based on game outcome
• ability to farm jungle creeps at will because the lifesteal keeps you full health
• allows you to stay on map longer giving you more xp and money and allows you to participate in more team fights
• Saves money buying health pots ( this might seem like a weak point but I personally buy like 2-3 health pots every other back when i go IE first. The cost builds up)

Cons:

• IE PD is a stronger 2 item power spike
• IE does more damage in team fights
• Crit chance is king for ADCS mid-late game

past mid game everything evens out because by that time you can go IE 3rd item and have the same build.

TL;DR (THIS IS COMPLETELY MY OPINION BTW)

• BT is a better 1 item power spike ( IE statistically only adds 100 damage every 5 hits. BT heals 100 damage every 5 hits PLUS has 160 hp shield at lvl 9 that scales to 350 at lvl 18)
• BT is a better item for lane ( Sustain = king in lane)
• BT is cheaper so you hit your 2 item power spike first
• BT allows you stay on the map longer (equating to more xp and cs you can get safely. It also allows you to participate in more team fights. it's just overall more efficient IN my opinon)
• BT allows you to farm jungle camps without losing health
• BT allows you to have more freedom to adapt to the game (you can choose between IE or LW 3rd item depending on how the game is going and the enemy's item build path)