A) Riot has never tested a cycle with its removal on the PBE.
But most likely internally tested it. As crappy as their internal testing is, that´s how the PBE even receives changes. Most if not all changes on PBE are passed through internal testing and approved.
B) Just as many, if not more, champions would be viable, even if the ones at the top would change completely. The ones that aren't at the top and get hit really hard by Flash's removal would stay exactly the same: unused and still needing Riot's attention.
Oh yeah, i would be so excited to see half the champion roster get shafted because of Flash's removal, regardless of said champions' usefulness in certain comps, which would either not exist anymore or be severely impaired.
C) Flash keeps strong champions stronger than weak ones because they make better use of Flash in most cases. I can't stress this one enough.
Flash also keeps League from becoming a shitfest of mobility by allowing immobile champions to actually get away. Because, let´s be real here, there´s no way in this universe an imombile champion is ever escaping death by a mobile champion, regardless of how perfect their positioning is or how safe they play. Riven can towerdive someone and get out on most cases. So can Yasuo, Irelia, Renekon, Pantheon. All of these and more would see considerably increased play due to how mobile they are and how poor towers are nowadays.
D) Flash's removal would be a smaller meta-shift than many pre-season/mid-season changes they've made.
It wouldn´t really. Mobile champions would cuck immobile champions even harder and with even less counterplay.
Towerdives would happen more often, especially with towers firing airsoft pellets at people plus tower first blood.
Mid would become the Assassin(and Ahri) Grand Central Station, resulting in Kayle and possibly Lissandra being played a lot more. And at Riot, Kayle seeing significant play means Kayle nerfs and Kayle nerfs mean she´ll stop being played and Kayle not being played results in total assassin(and Ahri) dominance of mid lane and the 'ADC in 2017 lul' thing becoming much less of a bad taste meme and more of a reality.
Kassadin may see play mid or top. Or mid and top.
Jungle would possibly change the least, with more Vi, Lee Sin and Hecarim and maybe even the surfacing of Riven into the jungle meta. Hard CC tanks may become needed to actually lock down all the highly mobile, juking bastards, although given the jungle's state, i wouldn´t count on it.
Tank supports with hard CC, i.e. Leona, Nautilus, etc may actually be useful in comparison to now to deal with all the mobility.
Top would see skirmishers and assassins a lot more, making tanks an absolute need in nearly all compositions.
On the other hand, Air Drake may stop being the 'Let´s get it just for RNG to, with any luck, shuffle in a different one' kind of drake.
In essence, we´d see hard CC tanks being used to stop mobility, mages falling out of favor almost completely, Kayle getting played to deal with assassins and consequentially nerfed, Kassadin actually being good, and the jungle staying on mostly the same old shitfest its been for the past season.
E) Only a few champions heavily rely on Flash. Riot tries using them as examples for why they shouldn't remove it when they are an extreme minority and would likely not be any worse off than many other champions in the game currently are.
Flash is either used to allow them to properly hit their rotation, or to simply not get ass blasted by a mobile champion. They would need to receive compensation buffs for the removal of Flash. And the mobile champions who, until those buffs, will ass blast them even harder will have to receive nerfs.
Just that Riot has never actually tested Flash's removal and every time they say its too ingrained in LoL's balance, they don't have any evidence to back that statement up.
Except it would bring a mobility meta no one wants. And an extremely weakened tank meta given jungle changes and not enough viable tank junglers.
It would result in a single extremely long PBE cycle, or several bimonthly ones, due to the fact Riot would need to analyze champion by champion to see who and what needs compensation buffs or nerfs.
Flash is too ingrained in League's balance nowadays. It´s something crucial at this point. Removing it would cause the game to enter a state of even more imbalance than it currently is.
I mean, Vayne is meta. She´s being played a lot, banned a lot and winning more than half the games she's played in.
League where Vayne is meta is League with SERIOUS balance issues. And no one wants more imbalance. There´s only so much people can withstand.
Were Flash removed before becoming borderline mandatory on every champion, and for some, absolutely mandatory, there wouldn´t be issues with removing it.
But as it stands, it´s too necessary to ensure League's questionably slim, but very much existant balance.
Its removal would do more evil than good.
Think release Syndra winrate for around a third of the champion roster, for most if not an entire year. Another third would be walking on thin ice, and the final third would be skyrocketed to quite likely 60%+ winrates.
An exaggeration? Perhaps. We´ll never really know what would truly happen unless Flash really gets removed. Presently, it´s a summoner spell that allows immobility to not be such of an buff warranting issue, and mobility to not become nerf warranting excessive.
Think of the amount of changes just removing Flash would cause.
Towers, items, jungle, champions, masteries, runes. It would be like redeveloping League from the ground up. While that may certainly come with its pros, i.e. less "itallian influence" in the code, the cons unfortunately heavily outweigh even the most valuable of pros.
TL;DR:
Is Flash a balance staple to League? Yes.
Is League better with it? No, i wouldn´t say it´s better.
But i do say it´s a necessary evil to, in more ways than one, make it so that balance is more a fine tuning, less throw it away and build it again from zero kind of ordeal.