Wait til you get mayhem. I’ve had insane drops on that alone otherwise during my time legendaries came from quests of the occasional boss
So legendary guns actually exist? 
Beat the game
Turn on Mayhem Mode
Do bosses that have spawn points next to them
Profit
I actually use the other characters as mules bc I have so many legendaries. Hopefully after week or so I can start trading them on the forum like B2.
That’s kind of the point. They give a steady stream of quest rewards so you always have at least 1 decent gun to progress through the story with. When you’re having difficulty with a boss or mob encounter, you can farm those guns and use them to get by until something better comes along.
And the quest rewards in this game are not nearly as powerful as BL2, where endgame loadouts were mostly blues (pimpernel, fibber, sandhawk, etc…) The quest rewards here are nice, but they don’t completely invalidate anything else like the Pimp and Sandhawk did.
I kinda see the OP’s point, but I’m quite happy having to actually sell or throw good gear away every 2 or 3 levels because I’m finding so much good loot as I progress.
And that’s a GOOD thing 
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Eh, I did get nothing of legendary weapons so far. But judging by the leg grenade mod I have which is making the game super easy, I am afraid of the consequence. I am starting to understand why people said the game is not challenging.
i’m on level 20 and i’ve gotten two legendary weapons so far and it was from Shiv so early in the game and a legendary hunt in Athena… i did almost all of the typhon logs even the one in Lectra city but still got nothing haha. not even from the golden chest 
but i got the walking gun (The Boo+) from side story mission (from Ava) and it’s been my go-to weapon so far
Last night I got 2 legendary drops from 2 different bosses, a nice corrosive SMG and a Flakker Shotgun (this thing is awesome!)
Tonight from one boss got a Legendary Rocket Launcher, a Legendary grenade Mod called “Moxxi’s Bouncing Pair” (love the humor in this one … and it is a great grenade), and a Purple Rocket Launcher that looks better than the Legendary one. This one has an underbarrel single rocket for 948 damage!
At first I was getting skunked but the last two days have been good for loot. Level 23 Fl4k in Normal Mode. Still working through the story.
I have to admit there are alot of Green and Blue weapons I have found that were excellent.
I’m currently lvl 31 atm and have approximately 3 legendary drops from my 25+ hours of play. One came from a vault boss, another was a jakobs sniper from an eden 6 quest. Other than that, nothing. Half the time even bosses just drop a measly couple of greens. Zer0 suit Kamigawa and GenIVIV to name a few. Hell, even Grave Ward from the second vault gave me a big old pile of nothing.
Absence of mules requires generous drop rates.
I wonder if the drop rate decreases as you level higher. Like, I’m having a blast with my gunner, paying attention to the story and learning her skill set. It’s fun! But. If the drop rate doesn’t decrease for TVHM or UVHM, then this game will end up losing interest for me, because I like to figure out tactics to grind for that rare legendary with sub-optimal gear. I mean isn’t that why we all play BL over and over and over? 
I have been playing since Friday and so far I have only one Legendary. But, it is the ‘Breath of the Dying’ from Dahl. Man, that is a sweet gun. I love it.
The game is amazing with so many quality of life improvements.
I do not mind how many legendaries dropp at all, it sort of reminds me of some hack and slash games where lots dropped, but only a few where good or something you wanted to keep, and got a feeling this is what might happen at higher levels also, but not there yet.
All you need is a multiple projectile torque shotgun used in sticky mode. Most of the Legendaries pale in comparison to a decently rolled one of those.
I am level 38 now, still not done with the main story and, I kid you not, the best “overall” weapon, that I am still using, is a Blue “Interloper” Shotgun 
I can’t remember, if it was a mission reward or a random drop, but that thing is absolutely insane. Ofc, there are other weapons that have higher “per hit” damage and are also fun, but this “shotgun”, that fires a single slug at high speed and straight, has high damage and VERY high radiation dot.
But there are many guns, that are just as useful, it’s just that radiation aoe dot effect, that I love so much ^^
edit: the shotgun is level 27 ^^
I’m seeing a lot more purple and gold drops than previous games, but the stats on these drops are not much better then green and blues, and sometimes they are worse when compared to the same item of lesser quality of the same or even lower level, I have gotten gold drops with no additional stats at all not sure what happened there but it should have been a white drop, then again I have gotten white/green drops that had too many additional stat and should have been gold/purple drops…
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Well my BLUE shotgun beats everybody. EVERYBODY!! It’s TEDIORE shotgun that have 5x MIRV that seek to enemies. And another TEDIORE SMG that have too 5x MIRV but no seek
You should certainly wait to open the chest anymore. The guns scale to your level when you open that chest. Saving it for max level, you will maximize your chance at the best loot from the chest.
Opening up before will be seen as a waste as the keys are very rare.
None of my quest items have been remotely close to the items I have found from things like loot tinks or even vendors when I level up a few times.
It might just be the character you’re using or differences in what others value verses what I value. Maybe some of these guys just have way better luck than me. But I used mostly blue and purple pistols and shotguns for the majority of the time leveling and ended up vendoring it all upon hitting 50. I think I only used 2 legendary weapons the whole time from 1-50 and never got any other legendary gear except a shield that didn’t suite my build. It wasn’t until I hit 50 and farmed that I had any issues with bank space. And now that I have a better grasp on what is “good” / “rare” I plan to clear a lot of that out too. I think there’s sufficient space, but if you guys do want to stash stuff, there’s always the option of making alt characters mules.