They can totally mute the flamer. I encourage everybody to mute the flamer, if they do not like it.
On the report part it is questionable for me. The flamer did nothing bad. He flamed and he got muted. And NOTHING happened.
"so that nobody else has to deal with the flamer" - You write this, like this is something huge to do. Pressing one button. It's not even "dealing" with somebody. It is just nothing.
It's like you want to punish the support, who got caught once, because the adc HAS to press heal if he does not want his support to die. It's not even like flaming, a support gettin' caught is much worse, because you have to waste a CD and it influences the game. More like if you want to punish somebody for asking if the jungler will build solari or the support should, because they have to react to this. If they do not react, they might end up with 2 solari, which is not a huge problem, but still.
Also chat related punishments are waaay exaggerated. Since everybody has the ability to stop you if they do not like you, there should be no punishment at all. (At max an automated text at the start of the game, that ***** player tends to be toxic, feel free to mute him now), or chat banning the flamer (which is already too much imho). But these punishments eventually lead to 14day and permanent bans, what is totally stupid. It's like killing somebody instead of asking for a paper which restricts somebody to go close to your house or schools or whatever ( i dont remember the name of this paper....)