I think the main reason why the game snowballs is the item system/gold system. Since now you can get tower first blood, you can get an advantage over the enemy team via gold which allows you to buy items that they can't get yet. Champions can't play properly without their builds, so of course if you get ahead gold wise you will tend to snowball if you keep that lead. Without key items, champions will get out dueled easily by those that do have them, thus creates snowballing.
Which in turn is why there's all these changes. And it's probably why the games on average take more than 30 minutes. Because teams try to even out that gold lead by stalling. It's a good strategy, but makes the game take quite awhile to catch up to the early lead.
If we look at another MOBA of the same style that doesn't rely on items, we see something different. Heroes of the Storm is very similar, except it focuses heavily on objective control and hero abilities. You don't need items, so you don't need to cs or farm for gold. The most you can get ahead is by level, but even then it's not that drastic. Kills are less of a focus in the game, and it's more about pushing lanes and controlling the major objectives on the map. Even minor objectives take priority over getting kills. Matches usually end in 20 minutes from my experience because of this.
I think if league didn't rely so heavily on gold and items, then it'd become less of a feast or famine type of game. However that's probably never going to happen, so we're kind of stuck with that play style :P
As for the dragons, I think they added them to add more value to dragon instead of being just another gold boost. Same with Baron and Rift Herald. Riot wanted to add more "objectives" to the game to reduce the need for snowballing, but really it just added to the snowballing. The only buff worth getting when behind is baron because of the pushing power, but if you don't have the items to back up the push you'll end up dying regardless :P
Burst and mobility aren't much of an issue. Champions were created with those specific characteristics in mind, so it's not that bad of a design.
The tower change was technically a good one, because for awhile towers took forever to take down due to their bonus MR and Armor. Not to mention the second lane tower used to give a huge shield, which made split-pushing and diving a nightmare. I don't think the towers are too bad, I just think they need better targeting AIs. More abilities need to cause aggro, and I believe Riot is currently working on that. Though some cases aren't exactly the best (See Yi alpha strike issue).
But yeah, to sum it up the game is heavily snowball in play style because of the huge need for gold to get items. Without items you can't do anything once the enemy gets some kind of lead. And with all these new ways of getting leads (tower first blood, first blood, rift herald, dragon buffs, etc), it just adds to the downward spiral of feast of famine playstyle.
I hope that makes sense :) Obviously I might be wrong on some things. But that's what I've been able to figure out using comparisons to other similar games.