I was also thinking anivia might be worth trying to get good with because egg sort of defeats the purpose of most assassins. Having to kill her twice might buy enough time for your jungler to come in and make a play.
While that's true in teamfights, that's not really true in-lane.
The only assassin that Anivia does really well against is Leblanc and that's only when your egg is up. If you fall behind and don't start stacking your RoA by 14 minutes, she can pretty much combo you to egg and then kill you for realsies 45 seconds later. Similarily, AD Assassins have no difficulty auto-attacking an egg to death. Fizz is a difficult matchup unless the Fizz is really bad (then it's mega easy). Cass, Azir and Syndra are extremely difficult matchups because you aren't allowed autoattack because you get "stunned" long enough that it's trivial for them to Q you.
She does have a very strong matchups against Swain, Vlad, Yasuo and Lissandra though.
She excels against hard-engage comps with tanky junglers... stuff like Udyr, Amumu, Maokai, since you get to just wall them off. Those champs just aren't played as much anymore since junglers are more often damagey and top laners go TP + splitpush.
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The thing is that you like to play long-medium range mages that are "lane bullies" that farm well ALSO scale well into the late game thru large amounts of AoE damage (Ziggs, Gragas, Brand are your big three that fit that classification. GP is another example of a medium-range lane bully you like to play).
In Season 2, champs like Karthus, Orianna, and Anivia "did it all" and were just as "OP" as you claim assassins are - they did exactly what the meta dictated. They were able to actually pick "Kills, CS Lead, Map Presence" because Map Presence didn't really matter until 20 minutes. Remember that people used to intentionally NOT take towers until 15-20 minutes into the game just because laning was so drawn-out.
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And i sort of disagree with your current "snowball" view of the current metagame. i feel like that is only true botlane.
While i disagree with Riot's current balancing approach to botlane (it's been extremely aggressive ever since Lucian was released, except that brief stint where Kog + Twitch + Tristana dominated everyone for about 6 weeks, and i feel like ADCs should be more mid-late game focused rather than early-game), the current meta is pretty decisively balanced around the mid-game.
Look at the past for snowballing:
Season 2 was the "farm for 40 minutes" meta. Champs like Karthus, Anivia, Orianna were king.
Season 3 was the real "snowball" meta. Assassins were actually good for the first time (as a class - not a single overtuned assassin like release Ahri or release Fizz). Do y'all remember how silly Oracle-Monkies were? Junglers literally only got boots + oracles the entire game. Ziggs was a 55-58% winrate champion for roughly 6 months.
Season 4 started off with their extremely silly 4-1-0 lane matchups in competitive play. That was quickly patched out. The game still had a huge snowball off of inhibitors/baron, but... generally you need to win a teamfight to get either of those objects. Teleport became common-place and teleport is intrinsically "anti-snowball" when you compare it to Ignite. There was the brief "hypercarry" patch which was 100% teamfight focused. Toplane Ryze was popular off-and-on thru the season, which is very much a teamfight champ. When they reworked Exhaust, Assassins were temporarily almost completely removed from the game.
The big reason that Season 5 started off Assassin heavy was exactly what i mathed out back with the first preseason patch:
1. everyone is ~30% tankier levels 1-2. Which is when asassins struggled.
2. everyone is ~7-9% squishier level 6-9. Which is when assassins spike.
3. Athenes nerfs + Morello buffs means most midlaners are going to have -20 MR compared to before by level 9
4. Botlane got super-aggro again so supports are taking Ignite more often => no exhaust for fed assassins or your botlane loses lane
(Fizz in particular got the strange MP5 buff that was out-of-place and was actually what pushed him over the edge).
And it's hard to say that preason-5 was really truly only "snowball" when scaling-teamfight focused champs such as Rumble, Lissandra, Gnar and Irelia were top-picks toplane.
And while assassins have more relative spike damage, junglers can no longer perma-camp midlane due to the increased difficulty of the jungle and the reduced amount of sustain. Which reduces the ability for a midlane assassin to snowball off of 2 easy 2v1 kills. And a bunch of assassins just got nerfed (with even more such as Katarina next patch), so assassins might not even be good in Season 5 - just 5-preseason.
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i feel like in-general you just go into frenzy the moment a particular champion icon appears. This goes BOTH directions btw: you get super excited the moment you see that TF lock-in and you get to go GP. It's not just Ahri/Fizz.
i feel like you put a huge amount of emphasis in matchups and counterpicks. (Espiecially when you don't have a lot of expierence with the matchup. Eg, Fizz is one of the classic counters to Ahri in competitive play, but you feel like Ahri beats Fizz because she DOES if you are not as expierenced on him - just like how Fizz can lose a lot of lanes if you aren't experienced on him).
We aren't at a Master-Tier level where it becomes harder and harder to just "out-skill" your opponent thru a skill differential and matchups become actually "really important". Sometimes you go against a mid-Riven as mid-Renekton and get completely demolished. Or you play GP-mid against a TF-mid, get camped and then the TF just ults everywhere and kills everyone else.
It's also kind of rude in other situations such as wheni only did well against Leblanc that one game because "Anivia trashes Leblanc". i was apparently just on autopilot and my champion just plays itself (and it had nothing to do with, for example, the Leblanc being lower skill and the Shaco completely neglecting my lane while Karl camped me)
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The 100% valid complaint you can make about ALL assassins is that: They tend to warp the entire game around their skill versus their opposition's skill.
This is just going to be a feature of high-mobility, high-damage, high-scaling champions, however. If you want to be able "carry a game", then you have to accept that certain roles/champions are going to be able to carry harder if things go their way.
