Well, there are several problems with assassins that Riot has consistently refused to address since their introduction to the game:
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Virtually none of them are significantly limited by any form of resources. About a third of them are resourceless or use energy (which is basically under an infinite blue buff). The few that do have resources have much cheaper spells than their counterparts in other roles. Talon can 1-shot a wave for 75 mana at max Rake rank, 90 for Ekko's Timewinder (who gets mana back from Dorans, so it's actually 76). The mages designed to actually clear waves pay more mana at level 1 than Assassins do at level 9 and their first back is dictated by their limited mana pool, which is heavily affected if denied any farm. Lux? 70/85/100/115/130. VIktor? 70/80/90/100/110. Ziggs? He needs at least 2 spells to clear the entire game, going anywhere from 120-180 mana. Assassins utterly lack that weakness in the hands of anybody above silver 5 who's not a support or jg 1-trick that doesn't know how to lane.
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They have the best wave clear of any class. Zed can infinitely spam his star-throwing shadowcopter, destroying every wave, every time until he decides to roam while poking you from literally 1600 units away - because energy. Talon can just buy 
, wait 3 seconds for the fighter minions to take a tiny sliver of damage and then 1-shot the whole wave with Rake. Ekko can do the same with a timewinder. Ahri does it with her Q and, thanks to her speed buff from it and passive sustain, can pick totally screw it up and not get punished for it.
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Melee assassins are better ranged casters than champs designed to be ranged casters, which lets them have the best of both bonus stats for being "melee" and the great clearing, safety and farming ability of ranged champions. They basically end up as hyper-mobile mages with bonus MR, more burst and lower cooldowns. Then, because of their heavy advantages loaded directly into their base kit, they get immense latitude to control the lane because you have to stay away from them basically from level 3 until the end of the game because they can all-in and kill you any time. With Riven, it's actually level 1. About the only exceptions to this are Kassadin and Akali, who wait until 6 but gain even more advantage than the others.
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**AD assassin builds are 100% unpunishable in lane, AP assassin builds only slightly less so.(( Oh, you killed Talon in lane at level 3 when he tried to all-in you? Great job, go get your...
for it's whopping 25 AP, which makes 40 AP with the 
you started with. Too bad for you Talon showed up with and can now kill the wave in 1 spell and utterly destroy you because he has double the MR that you have armor. If he's really bad and the lane is still difficult, it's ridiculously easy to just build 
and now no AP champ against him even has enough mana to kill him. No matter how badly they mess up, their next item is only some low multiple of 350 gold so forcing them out of lane has no negative consequences. Even an AP champ gains a huge spike with a single amp tome they can build into Morello or aether wisp. But if you're a regular mage, God help you if you die more than 30g short of the 1250 you need for an NLR or 1st augment for Viktor. If you miss it, you're basically screwed and you'll lose at least 2 waves recalling early to get it.
5a. AD assassin item builds grant obscene amounts of "extras" for no trade-off. AD assassins can build 
and get: +60 movement speed out of combat, +45 movement speed all the time, Tenacity, a spammable spell shield that doesn't punish them when it's interrupted (with a very short 45-sec cd), +220 Bonus AD (more even than ADCs build), +30% CDR, +400 hp, constant effect ward sweeping and blackout until triggered (only 90-sec cd), an instant 20% speed boost (only 45-sec cd) a true damage nuke, +70 MR, +45 Lethality, 30% armor reduction (which they can all apply as AoE) and there's still another item slot open. Since Assassins don't actually need more than 3 damage items to kill a non-tank >1 second, they get the choice of more defensive stats and a free revive with 
, more cc/spell immunity and MR with 
or 
, AD/lifesteal/% damage mitigation/CDR with 
or even more armor negation with 
so they can assassinate tanks. And they can do it all by building straight top-tier damage. An AP champ couldn't even get half of those free abilities and the few they can get, they usually have to sacrifice 40-65 AP a piece for when lost deathcap scaling is accounted for.
5b. Runes, masteries and stats disproportionately favor assassins even more, especially AD assassins. Back around the start of S5, Riot decided to change the way champions gained stats per level by changing from flat per-level values to base stats and a growth stat formula. This left champions universally lower HP and armor in the early levels, which is a total godsend if you're an assassin, because your opponents are all so squishy you can all-in and kill them at level 2 or 3. Then Riot got the idea of adding Thunderlord's Decree and giving them even more burst to delete your already-lowered stats without generally reducing assassins damage output. Now lets talk runes: marks - best at physical damage and penetration stats, Quints - great at everything, seals - best at resource regen, hp and armor, glyphs - best at AP and MR. Notice anything funny? AD assassins don't need any mana regen or magic anything at all, so they are free to take the best stats from every rune type. Every. Single. One. An AD assassin can walk into lane with +24 lethality, +9 armor or +72 HP, +13 MR. They walk into lane at level 1 - at their weakest point - and they already have a massive advantage. For free.
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Mobility spells are just flat-out unfair and Riot is only making them even more broken. Riven? SHIELDQUADRADASHDOUBLESTUN and you're safe. Talon? Hop 4 walls. Zed? Blink back to your ult shadow or throw one over a wall and switch. LB? 2 dashes and 2 blinks. Kat? You have a blink every >1 second if you touch a knife. But if you're a non-tank non-mobility champ, you get one mistake every 5 minutes. If you're Xerath and you overextend, you die. Add 1 to the second score column and get ready for your team to flame you. If you're Brand and you make a tiny error in positioning, you're dead. Varus? Dead. If you get engaged on by an assassin and manage to flash away, they'll get their ult up at least 3 times before you get your 
back. They can all-in you over and over, with no significant risk and they have to commit nothing they won't get back very very quickly. Oh, and all those assassins have the same flash as you, they just don't need it to escape so they can match you flash-for-flash on all-ins. So your only escape isn't one.
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Most assassins don't need to snowball anymore. The strongest AP champs late game are 
and 
. Notice anything weird? How about the fact that there are 8 god-tier late game APs and * half of them are assassins**. And they're the best, even at teamfighting. They have obscene DPS and, unlike the mages, they also have the option of splitpushing. Let's look at AD late game champs: 
are the physical damage dealers that really go bump in the night if you let them get to late game. Yas, Riven and Vayne are all "Slayers", Riot's new buzzword for "assassins that also blow up tanks". 3/6 are assassins, and not only are they pretty damn good in team fights, but they're among the best split-pushers in the game. In terms of splitting, you can even add Talon and Zed.
All that stuff right there? That's the problem.
