It definitely is a chore but I find it to be worthwhile, on that vein of logic why even require gold or any items to drop at all, picking them up and finding them is a chore
As Syl said, it makes those epic items much more epic because you have something really terrible to compare them too. Nothing says you *have* to pick up every single white piece either and in fact, I only pick up gear I can use and stuff like potions (unless I'm strapped for gold) and ignore everything else. Not to mention the lowest tier of items will *always* be trash items regardless of the level of it, so if you remove whites the next tier up would become equally useless to you and be nothing but vendor loot.
Well, in TL thanks to pet it's not a lot to do, just few clicks, only downside is "pet downtime", when I play for example path of exile i mostly ignore white loot.
I don't mind white loot dropping, i just want easy way to get rid of it. And that depends on game's "gold level", if I can't do anything with excess gold I dont care but in TL there was OP echanting to spend gold on. Even something like "sell now but for 3/4 of shop's price" so you can either sell by pet, get full price or pet downtime, or have pet always with you but get less gold from it.
Loot drops in aRPGs will never make sense anyway
("this goblin dropped 2h sword ? where did he put it and why he was hitting me with rusty dagger instead?").
Also, anything above white have some kind of random enchants or gem slots so it could potentially be useful (as green/bluedrops can get random rolled in a way it doesn't fit your build and with greens you have more "rolls" to hit combination that's best for you).
Other way to deal with white gear (instead of "sell or ignore") would be some kind of crafting (disenchanting/salvaging) that gives mats from which you can make some randomized item with chance for better q one. Or to make item that can be used on green/blue drops to "reroll" stats, maybe with small chance of boosting quality or adding a gem. I think D3 wanted to do it but they removed that in fear of devaluating crafting materials on AH