I concur that flagging an entire game for review (instead of reporting a specific player) would be better for reducing toxicity. But I don't think volume of reports is really a problem, Riot seems to handle the quantity of reports fairly consistently already. Perhaps if it would benefit the accuracy and efficiency of that system then it would be good to try to reduce the volume of reports, but I don't see it having that much of an impact.
I think it would do a lot to help if Riot made the punishment system feel more fair to the players. For example... In the current system a person can have a clean record, and receive a 14-day ban for using zero tolerance phrases. This in and of itself is not a problem, players should be justly punished for that sort of thing.
The problem however, arises when that player comes off of their 14-day ban. They understand that zero tolerance phrases are not allowed, so they stop using them entirely. This is good, and the player has genuinely made an effort to improve their behavior.
The only problem is, they are still sometimes negative without realizing that this is also considered toxic. So then they trigger the punishment system again, but the next step in punishment is always to escalate from one tier of punishment to the next, and the next step after 14-days is a permanent ban. So they receive a permanent ban the first time they are punished solely for being negative in chat.
I believe this problem could be solved if each type of behavior had it's own ladder of punishment, so that a player is given a fair chance to stop each behavior before receiving a permanent ban. More severe behaviors (such as int feeding or using hate speech) could still skip the earlier tiers of punishment and result in a 14-day ban, but this would not automatically put the player on probation for other behaviors.