Limited Summoners per team

So this idea just came to me seeing all the semi-hate around balance and prevalence of certain summoner spells. What if each team, collectively, could only have 2 of any one summoner spell across the whole team? (Like 2 ignites across the team, or 2 TPs, etc)

Before you all instantly hate on limiting anything (like flash, by far the most popular summoner out there) think about all the ways this could open up gameplay. What if you could only take 2 flashes per team? That opens up at least 3 slots out of 10 for other spells. Things become more fluid - perhaps a double heal bot lane works really well, or maybe a midlaner with tp and ignite (which we sometimes see already). It also impacts how champs play and could lead to a larger understanding of champ power vs summoner power - flash tibbers is pretty powerful, but what if annie doesn't have access to flash? Does she then need a buff considering she won't have (the option of taking) flash every game?

Of course, realistically, this just gives one more thing for people to argue about in the pre-game lobby. The amount of communication required would probably be beyond what any non-professional team could sort out in that 90 seconds or so. So I don't expect this to be an idea that ever really gets in, but I thought it would be interesting to share it and see if people could look past not having flash every game to seeing what sort of interesting play patterns this might generate.