The issue is the ease of getting FNP, and at little to no cost.
First off, FNP is the first thing you gain from Power from Pain. I think the chart is upside down, or at least mixed up. FNP is clearly the most powerful of the abilities, yet the easiest to access, meanwhile fearless is the last.
The chart should be-
1- Furious Charge
2- Fearless
3- Feel No Pain
The concept of DE is to be a Boxer with a glass jaw, but gaining FNP increases their survivability by light years. Furious Charge and fearless are more aggressive rather than defensive abilities and reinforce this concept.
Sure S6 weapons can get around it, but no one walks around with basic S6 and armies generally have to pay a decent amount to gain them (plasma guns are 15 pts, 35 pts for Assault cannons on razorbacks).
Then there is the ability to just hand out Pain Tokens right off the bat. Buy 3 Haemonculus. That is 3 pain tokens right there for 150 pts. That is 3 units with FNP. Okay before you say it, I know that is the same as a Sanguinary priest in BA, who gives Furious Charge and FNP. The difference is that the Sang Priest can be singled out, killed in combat, and then my Blood Angels loose the ability. The Dark Eldar though keep it, even if the Haemonculus is dead. I will say looking at it this way you can make the chart this
1- FC
2- FNP
3- Fearless
BUT FNP should still not be first.
Then there is the Duke, who doubles your chances of getting a 6 on the combat drug table, allowing units with Combat Drugs to start with pain tokens. A 1 in 6 to a 1 in 3 chance is a big leap and a great improvement.
Another thing is that it is easy for some of the units to get FNP turn one/two with a one sweep assault, especially against armies like guard, tau, orks, and necrons, and nids. Arrive and get to assault asap, minimizing time you are shot at, and then get to render small arms fire insignificant the rest of the game.
As well handing out FNP like this makes armies that had it before less unique and competitive. I even feel this way about BA somewhat, though I still utilize it, as will all the Dark Eldar. But things like Death Company in old BA, and Plague Marines in CSM lose some of their inherent coolness and advantages with the onset of the design studio's mindset to hand out USRs like candy.
So what tried to do is design an army that hits really hard and really fast with the intention of making them fragile and easy to break. What they did was make an army that hits hard and fast, but is resilient and has staying power as fast (or faster) than its ability to deliver the hurt. It just doesn't make sense as the first ability and makes the army more resilient than I believe it was meant to be.
EDIT 5 NOV
In response to a lot of the comments. Yes I do understand that the army can get stranded and shredded if you shoot the transports, and yes I know S6 bypasses their FNP.
The issue is, they have defenses against this as well. They are a very fast Army. Their transports turn one should get a cover save because they should turbo boost. Turn 2, they should be disembarking and assault. This really minimizes the time I have to shoot at them before they gain FNP.
I think a better mechanic would have been as follows:
Darkness Prevails: Dark Eldar always have +1 to their cover save. They gain a 6+ cover save in the open.
To me, this is more fluffy and a little less powerful, lending to the Boxer with a Glass jaw concept.
As for other FNP armies. Nids can get it easily but don't have the Speed of DE/BA and the Mech of CSM.
BA have Sang priests yes. I will concede that this made my very own codex a little bit broken. They are virtually flying PMs. But there is the other side to this. SPs are harder to utilize and are vulnerable AND Space Marines don't benefit from FNP as much as DE do. SMs already have 3+ saves, so you get a marginal benefit from FNP, as you don't fail too many to begin with. DE on the other hand against a bolter are going from 0% chance of surviving a Bolter to 50% chance of surviving a Bolter. And cover saves are abundant in 5e, so you need to account for that as well.
As for PMs, they used to be the FNP kings. Now, they are more generic. While still good, they are pricey troops whose advantages can be found elsewhere in the game. Plus, overall the CSM codex is shunned as a lackluster codex in need of revamp. Plus they can not gain the other abilities that DE/BA can gain, and so their advantages are even surpassed.
You can say, as the BOLS post did that CSM have yet to be updated and we will see new advantages, but there are 2 problems with this IMO: When will that update come and then there is Codex Creep.
As for S6 getting past it. S6 isn't that common and it is expensive. Yes, people can spam it, but for a balanced list to face all comers, you won't. So to me this isn't as big of a disadvantage as people claim. In general, you will find more S8 weapons in an army list than S6.
BUT know this: This is a rant. It is a slight exaggeration of my true feelings and everything across the internet comes out 10 times more dickish than it truly is. The DE codex isn't the end all be all of cheesyness and beardiness like Necrons were at first release. IG are still top dog IMO. I just feel that DE are overtough for what they were intended/should have been. That is not to say that they are the toughest SOBs out there and no one can topple them. I just feel that the Power from Pain rule went one step too far.