There's a lot to debate here, but I'll try to cover some of my own personal points and opinions on the matter.
Flash is a near mandatory skill because it allows play making and provides all heroes with both an offensive and defensive ability that is frequently game changing. In this regards, it could be considered terrible for the game from a game design perspective. It is singularly so impactful to game play that all heroes must be designed and balanced around it's existence. However, from a player perspective, Flash is actually acceptable, or even desired. To understand why, we can look at Dota, and other Mobas. It is a fairly uncontestable statement that heroes in LoL counter each other to a far lesser degree than heroes in Dota. Why is this? One of the major factors is the existence of Masteries, Runes, and Summoner Spells. Each of these serve to allow individual heroes to adapt against other heroes. In this case, Flash is ESPECIALLY effective - can you imagine how heroes without an escape could possible escape a J4 Ultimate? They cannot. Thus without Flash, if one team saw their opponents pick non-mobility heavy composition, then picking counters such as J4 who can easily and continuously abuse that deficiency would make for a much more one sided game then if Flash were an option. This of course leads to the question of: Doesn't Flash therefore reduce the strategy involved in pick/ban phase of the game? In short, yes, yes it does. Having played other Mobas for 4+ years now, I can tell you that team compositions and lane matchups matter far and above more at non-competitive level game play in other Mobas like Dota versus in LoL.
tl;dr - Flash is mandatory because how uniquely powerful for its offensive and defensive uses, allowing it to single handedly change how games are played and won. While un-elegant, this isn't necessarily bad because it allows more freedom in team composition, though at the cost of some strategic depth and hero diversity.
Ignite can be seen as the counter to Flash. How many times do you notice that you flash even though you will die to ignite? Instead of trying to chase enemies for that last bit of damage, (after they escaped using flash, or some other permissible mobility skill due to the existence of Flash), you can simply Ignite the opponent. On top of its extra burst damage and occasionally useful healing reduction, I think it is this factor that makes Ignite the almost de-facto second summoner.
All other summoners except revive seem to have their own situational uses that make them acceptable, and I don't think there is any complaints there.
is always being used and a secondary spell is usually 
. Of course junglers use 
and supports use 
unless they have enough CC to use 
but of course 
and rarely 
.
{{summoner:2}} 
all could use some more or less small buffs:
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