As an individual who was around during the days of the Tribunal, let me start by saying that player behavior and reform has come a long way.

That being said, there is still more work to be done.

I believe the following additions would be good for the punishment system:

1. Text in-game intervention to catch behaviors and warn players
2. Alter the in-game music and audio during death

These suggestions come from my personal belief that most people who get punished are NOT acting inappropriately due to deriving pleasure from their misbehavior when they're aware it's wrong, but rather, individuals who get lost in their emotions during the game and forget how to interact with people due to the lack of face-to-face interaction.

In-Game Warnings

Currently we have a system in League that can go through chat logs and identify if the reports received for an individual correspond with their text chat. On the back end, Riot has loads of data that can be analyzed to identify words that correspond to reportable behaviors.

I don't see why this cannot be instituted on a smaller scale during a match. Such implementations would allow for live, real-time interventions regarding behavior. The sort of messages would be on how to relax (breathing techniques, coping mechanisms, etc.), how to refocus on the match itself rather than how team mates are doing, and/or straight up informing the player that they are behaving in a way that might upset their teammates and they need to reconsider their actions. I do not believe this sort of analysis would be terribly taxing on the current systems in place, but if need be, it could be instituted on players after receiving n amount of reports and stopped after k number of report-free games.

On second thought, informing players how they're misbehaving while they're playing might distract them, but the other suggestions - relaxation techniques, refocusing - shouldn't have a negative effect. Refer to the source on reducing adolescents' aggressive and hostile behaviors.

Alter the in-game music and audio during death

This might seem like a strange one, but hear me out. Music therapy is a thing, has been a thing for a long time, whether you are actively participating in it or not. During the death screen, when everything is gray, it might be as simple as slowing down the BPM of the music to produce a calming effect. Alternatively, different music altogether with a slower tempo might be better - the music that typically plays in game almost become white noise or otherwise unnoticeable, and the only way to have it have an effect on players would be to change it subtly such as slowing it down and/or making the music softer, or to play new music altogether. This might actually be good for the game as a whole, rather than just misbehaving players; it's less of a punishment, more of a way to make a player to calm down. Refer to all other sources for this one.

Sources

• Increase in salivary oxytocin and decrease in salivary cortisol after listening to relaxing slow-tempo and exciting fast-tempo music. Yuuki Ooishi, Conceptualization, Formal analysis, Investigation, Methodology, Project administration, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing, Hideo Mukai, Conceptualization, Formal analysis, Investigation, Methodology, Project administration, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing, Ken Watanabe, Formal analysis, Software, Suguru Kawato, Methodology, Resources, and Makio Kashino, Conceptualization, Funding acquisition, Project administration, Resources, Supervision, Visualization. Editor: Urs M. Nater. Article on PubMed: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5718605/ .
• Reducing adolescents' aggressive and hostile behaviors: randomized trial effects of a brief family intervention 4 years past baseline. Spoth RL, Redmond C, Shin C. Abstract on PubMed: https://www.ncbi.nlm.nih.gov/pubmed/11115311 .
• Significant treatment effect of adjunct music therapy to standard treatment on the positive, negative, and mood symptoms of schizophrenic patients: a meta-analysis. Ping-Tao Tseng, Yen-Wen Chen, Pao-Yen Lin, Kun-Yu Tu, Hung-Yu Wang, Yu-Shian Cheng, Yi-Chung Chang, Chih-Hua Chang, Weilun Chung, and Ching-Kuan Wu. Article on PubMed: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4728768/ .
• The Effect of Music on the Human Stress Response. Myriam V. Thoma, Roberto La Marca, Rebecca Brönnimann, Linda Finkel, Ulrike Ehlert, and Urs M. Nater. Editor: Robert L. Newton. Article on PubMed: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3734071/ .