To answer your first question shortly and sweetly; they maintain punishment systems because they know the playerbase doesn't log in just to get flamed. They log in to play the game and have a good time, and getting flamed is not conducive to a good time. It's not an SJW attitude or a waste of money - it's an effort to maintain the quality of their game for the playerbase.
The mute feature isn't an excuse to allow flaming, and vitriol is not an intrinsic part of competition, and since flaming serves no purpose but the short-lived satisfaction of the flamer, it's very easy to understand why it's not allowed.
As to your second series of questions;
Secondly, how does the punishment system work? Do you first get chat restricted? Then a 7 day ban? Then 14 days, permanent? Do they punish you by analyzing one game?
The IFS, when dealing with chat-related misbehavior, views the logs holistically using knowledge built up from what Riot fed its neural network. It reviews any game in which a player was reported, and decides, in a case-by-case basis, whether or not the reports filed against the player were valid or not.
It only requires at least one report to perform a review, and further reports do not increase the likelihood of a punishment. If the IFS decides to apply a punishment to a player, it will typically try to present up to three recent matches showing player misbehavior (in order of most egregious to least) as evidence supporting the punishment.
The basic escalation for the punishment system is as follows:
- 10-game Chat Restriction
- 25-game Chat Restriction
- 14-day Account Suspension
- Permanent Account Suspension
Assuming a player commits modest offenses with enough regularity, they will progress through those tiers from top to bottom. However, especially egregious offenses (hate speech, encouragements of self-harm, etc.) can warrant an escalation to a 14-day ban off the bat.
And, the IFS seldom punishes based off of one game alone. Generally speaking, it punishes based off of consistency and severity - multiple minor infractions will build up to a punishment, but infrequent, egregious infractions can lead to a faster punishment.
And, another worthwhile note; when the IFS reviews a player's behavior, that player's behavior is reviewed in isolation, as other people's behavior has no bearing on how the player decides to react.
If you've any further questions about the system, do feel free to ask.