No, you don't want to start by farming EE or legendary items. That doesn't allow you to level in heroics or epics and get past lives while farming gear because your character will be ungeared and too weak to do so. You want to first farm the heroic gear set(s) for your build(s). Then optimize a lvl 15-21 gear set around that heroic set, which will allow you to level in epics and legendary.
A cap alt is amazing for farming items while your other characters level.
That optimized set of level 15-21 gear may allow a well-built character on an up to date build to solo legendary reaper Sharn around level 21 (maybe EE or EH for weaker builds and characters), which is then when and how you get the epic gear sets over several epic lives.
Don't pike, don't beg for scraps. Assemble a
level 15-21 gear set of Death and Destruction, then go get your own gear as you do epic past lives. Any level 30 character, even a first life, should have no problem in heroic elite Sharn farming gear or collectibles for Cannith Crafting. The epic and legendary gear you really don't need and will barely use anyways for past lives since it is mostly level 29.
My characters did a half dozen epic past lives with a ton of Sharn before even having the legendary sets to use at 29. Which didn't bother me at all, because the foundation 15-21 gear set was solid, and a 29 set is barely used.
A lot of people try to skip steps. They try to skip to legendary without mastering the heroic and low epic item base, or they try to skip to reaper without mastering elite. Or they try to skip to heroic reaper Sharn without a solid heroic cannith/rl set. The game just doesn't work like that.