So here's the best reasoning I can think of all the complaints about assassins. Not sure if it's worth noting at this point but here goes. RIght now, at this moment, assassin's are the worst designed class in league. There are a lot of factors that go into this so let me put out the biggest ones.
Firstly, mobility. Mobility is a very frustrating thing in league because it gives a lot of freedom in any given situation. Assassin's power budgets never take mobility into account so they tend to very in their ability to get in and get out of a fight. No matter what assassin you talk about whether it is zed or fizz, akali or rengar, you will see that the ability of assassin's to get to their opponent/stay on their opponent is hated by most carry roles. This is further augmented by the second point.
Damage. The damage of assassin's generally isnt that great compared to mages. whether it is annie, azir, orianna, lux, corki, vayne, caitlyn etc, most mages and adc's as damage roles can outdamage assassin's. Then why is damage a problem? Because assassin's do it faster. It doesnt matter if azir can do 2x the damage of leblanc if leblanc blows him up before he does anything. Combining this with the aforementioned mobility gives every assassin by nature, decent early games a terrifying mid games. In lane, once an assassin has the ability to all in you, they can straight win the lane with equal skill just because they have low margins for error. You blow everything as an assassin and you will generally have enough damage to kill something because most mages/ adc's need time to do lots of damage or are balance around doing damage to an entire team. Because an assassin's damage is single target they can only delete one champion, but early and midgame are literally defined with single situations as more than just a probability, they are a fact. Until both teams start doing 5 man groups late game being an assassin is generally not that hard because at some point someone will always be isolated and SURPRISE QWER you're dead. This brings us into the 3rd problem
Scaling. As I mentioned before assassin's have strong early games by nature. Maybe you want to avoid this and say akali pre 6 but that's the point. Once she hits 6 akalli is plenty strong and can straight kill her laning. Add to this the fact that you have a few, leblanc, fizz and akali come to mind, who hyper scale into the late game to delete anything and viola you have a recipe for a champion class that will frustrate nearly everyone. It makes certain situations very binary where the equation is: if assassin is present you=dead, if an assassin not present you=alive. This is further augmented by the fourth major problem of this class.
Itemization. this is actually the worst part of the class. The way that league works, every class has its own items/group of items because the class wants those stats. ADC's want Crit, AS, AD, lifesteal. FIghters want health, resistance and damage. Pure tanks want just health and resistance. Etc for mages and supports. Not only do assassin's go down both AD and AP as their damage system, their are currently no items meant specifically for assassins. Think about it. All the items for assassins fall into other classes like ADC items/fighter items for AD assassins and mage items for AP assassins. The only stat both classes share as wanted is penetration which is still split between AD and AP. This means that they can fluctuate in and out viability much more randomly then any other class. The only explicit nerf to assassin's from an itemization stand point is when tanks are buffed. At least in theory, assassin itemization should be filled with low scaling, high penetration items to give them better early games and worse late games or something akin to this, but because this is not the case, all assassin's are allowed to just keep scaling to high heaven.
tl;dr there are 4 main points to all assassins that make them frustrating.
mobility because it can give safety and reliance.
damage because while not exceedingly high it is all single target and thus more than effective.
scaling because they basically become strong once they get access to both damage and their mobility which can happen as early as level 6.
itemization because it means any assassin can become relevant at almost any time.