Assassins can exist, just not in the way the game is now.
If you have ADCs doing 100-0 damage in 3-5 (2-3 seconds) autos plus a rotation of spells, fighters doing 100-0 damage in 4-5 seconds with sticking power, mages doing 100-0 damage in combos from range, and even tanks straight up being able to cc lock run you down for 10 seconds to kill you - then assassins won't be successful without 100-0 damage potential in one combo.
So, really you do need to change damage as a whole to make assassins allowed to exist without being broken.
If everything did half as much damage, we wouldn't really have a problem.
Assassins would never 100-0. They would be great at cleaning up targets that are at 50-60% health. THAT'S what an assassin SHOULD do. You're not straight up murdering someone who wasn't even positioned that badly. You're good at finishing off a low health target. Like people complain about Ahri being able to get kills with R-W and not needing Q or E, but really that's a good assassin playstyle. You barely got away with <30% health and the enemy team has an assassin in the area? Yes, you're dead. The assassin's skill was knowing where to go.
What's not okay is getting a small lead, then immediately 100-0'ing anyone that's not a tank if I land a single charm. However obviously Ahri needs to be able to do that with a lead, because otherwise she'd be useless when everyone else is doing it.
I don't think any champion should have a full combo doing 100% health damage.
Assassins: Should cap out at 60% damage when even, and can probably get up to 70-80% when fed.
Fighters: Should cap out around 40-50% damage when even, and getting ahead increases how long they can survive staying on your afterwards (as well as a bit more damage). Obviously they'd be beefier than assassins and their auto attacks should do more damage while they're sticking around.
ADCs: Can do like 20-25% with their spells, but mostly are focused on auto attacks and should need like 12+ to kill a normal target when even (maybe around 8 for a squishy target and 20-25 for a tank).
Mages: Can do 40-50% damage with a rotation when even, getting ahead can increase that to 50-60%.
Tanks: Can do 10-20% damage with a rotation. Getting ahead may increase slightly, but really they're locking down your damage for a long time.
You'd half any of those numbers if used on a tank.
Then there you go. You can have a fighter or mage successfully land a combo on the enemy carry, and THEN your assassin can go clean them up. And it would be okay, because the carry would have had to mess up to let your fighter/mage start that combo.
Tanks would actually survive for a long time (although honestly full tanks DO seem to be surviving a bit longer now).
Teamfights would actually take like 10-15+ seconds, not like 3-5 seconds.
, this was really when they started treating assassins as they do today, and they started converting champions that use to be considered mages or fighters 
under this new fantasy of "assassin"
nuking you from the other side of the map is fun
auto attacking everyone to death is also fun
(make it a defensive item only, aka revert it).
