If Cass was just too strong at all points in the game, nerfing would have been easy. The problem was that Old Cass was an oppressive laner that ended up having no point in the late game. The need to simultaneously add counterplay to her laning and also buff her lategame is what make the rework such a challenge.
I think you fail to comprehend the whole point of the role "lane bully."
That is the sole reason you pick lane bullies is to try to shut down a dangerous lane opponent in the hopes that your team can take advantage of that. You do the heavy lifting in the early game so that your team can close the deal instead of waiting out till the late game and do your hard carrying then. A great example of a lane bully that was extremely popular in the past was renekton.
The reason lane bullies can be bad for game health is IF they do have a strong late game, that is when they become over bearing and a game health problem. Falling off in dmg late game is a cost they take for having such a successful laning phase. And I feel that due to the lack of lane bullies assassins are having a heyday with it. (that, and zed's laning phase is way too safe/easy)
Reliable single target burst damage is not good for the game. I know Veigar players liked that. Doesn't matter. Can't support that fantasy in a competitive game.
I strongly disagree. I fully believe a single-target burst play style can not only be healthy for league, but that the distinct lack of such champs is actually hurting league in the long run.
Having a champ dedicated to deleting one target causes players to actually think out what they are doing rather than blindly jumping into a fight and hoping they land their stuff before jumping out again. It also adds to the importance of positioning both before and during team fights by forcing players to realize that their dashes and flashes won't always save them from being stupidly out of place.
I find the distinct lack of inevitability causes players to forget about planning and the importance of each choice they make over the course of the game can have a lasting impact on how the rest of the game plays out.
Veigar in particular did this well. If you let him farm in lane without mercilessly crushing him, he will turn the tables on you. Veigar's theme is about certain demise and unrelenting revenge. And that translated well with his reliable kit. He created a "death zone" around him that all but the tankiest of champs should be scared silly to approach.
That being said, Veigar was far from being a "broken" champ. I've had numerous games where I was so fed, I had more gold than I knew what to do with, but despite it all, I could not win the game simply because after 40-50 min into the game, there are just too many enemies that need to be killed. And I can only kill so many of them within a certain time period. It didn't matter if I could do 3k burst dmg to a single target if they all had more than 2.5k hp and 100 MR. It's even worse when they all buy both a banshee's and a GA which completely negates my usefulness until my team can burn through their special defenses.
There are numerous ways to over come single target burst champs, and I believe that over the years, riot has slowly been taking away the need to use these methods in favor of reactive plays and such.