Nami and Sona point out the issues with Lux Support glaringly.
She doesn't support. That is her problem.
Long CD's(10 seconds is a long CD when its your only form of peel, her shield is maxed last so is at a 14 second CD the majority of the game, and her E only gets down to 8 seconds and is spammable because its intended for wave clearing.)
Unreliable peel(snare blocked by minions, shield can be straight up missed/dodged out of by accident, E can be over shot and missed and doesn't actually slow by that much; 45% at max rank compared to someone like Braum(70%), Sona(40% at ALL LEVELS and scales with her AP), Nami(Ult starts with a base 50% slow and her E slow is small but also scales with her AP), and many others, this is honestly trash)
Low Base Damage and more scaling means that she does less damage than many of these real supports do due to the lack of AP and the lack of actual gold income
No abilities by which to actually assist the carry; no enchancing abilities or bonus damage or meaningful ways of protecting her carry.
Her primary poking capability actively pushes the wave and takes CS from the ADC, stealing the gold from the carry and screwing up any wave manipulation that the ADC was trying to have. This is an issue that many mage "supports" have as they are intended for this magical thing called a Solo Lane.
Her ult isn't helpful to a teamfight; its just a big, easily dodged AOE nuke that, once burned, just means that the enemy team can just carry on running your team down. It doesn't disengage, doesn't engage, it can't be used to peel or assist allies. its literally just damage and that's it.
Low mobility means that if she gets engaged on or fucks up her combo, oops, now she's about as useful as a cannon minion; she is a champion that needs peel. She does not peel.
If behind, she's fairly useless as her CC is, again, unreliable and her damage will be atrocious because support income is bar none the lowest in the game. She won't have the levels to have even higher base damage and she won't have the gold to have much needed AP items. And she won't be useful to her ADC or team either as she can't peel as affectively as a true support actually could.
She is squishy and leaves a much needed gap in the team as her skillshot shield can't even necessarily provide pseudo tankiness like other, more reliable supports(Nami's heal can negate burst if timed well, Janna's shield, Lulu's shield, Taric's shield, heal, ult and bonus Armor... All infinitely more reliable and easier to apply than her shield.)
While we're on the topic, the strength of her shield is grossly over estimated. On first cast, the only one that you're likely to actually hit reliably, it only gets up to 110; Lulu's gets to 210 and is guaranteed. Janna's gets up to 210, is guaranteed, and gives 40 AD. The initial shield is tiny and the second, larger part is only guaranteed on Lux herself; not her carry or her entire team. At best, she will land the first part on one person and the return part on some other part of her team and even then, its only 10 more than the shields provided by other supports who, again, are more reliable.