Only run DFT if you plan on doing glass cannon illaoi (my preferred build) or 3 dmg items. Anything else and Grasp should suffice for your needs.
As for core mine's cleaver+deaths dance. Cleaver should be first buy if you're behind or second if you're snowballing (in that case buy deaths dance first). As for beef, visage, dead mans, randuins should suffice
If you're popping with that much HP in team fights you should work on how you engage as the only counter to illaoi is crowd control. So you popping leads me to believe you may be getting off sub-average ults that don't heal you effectively
Or, you ult, get CC'd and end up dying before you can slam anyone
If you have spirit visage and land your ult on at least 3 targets (or 2 people+the spirit) they should heal you for a crap down of HP while dealing dmg and shredding armour (more than enough HP to survive the initial burst of a team fight)
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I use a FULL AD glass cannon gismo with my own twist used by a diamond illaoi on NA. Obviously I'm no where near his level but it works as long as you ban fiora.
My build is generally
into
into
into
then
///
for dmg or
////
for defence. Buy boots whenever they fit into your budget (sometimes I even sell my boots just for more damage).
If I fall behind I may grab a
but otherwise I never stop building damage.
With this, you can 1v1 almost any champ in the game, ganking you becomes a suicide mission for the enemy jungler and with a nice 500+ AD and a auto reset..... you're split push god.
Edit: found him https://na.op.gg/summoner/userName=Maulface