It doesn't do anything.
The system tracks the frequency of the games you are reported in; it doesn't care how many times you are reported in a single game.
Quite frankly, your report alone isn't treated any differently than 9 reports in a single game.
The only thing asking for reports does is ruin team morale even further.
Riot employees don't threaten to report people or harass them regarding it. Considering they are the final authority in the matter, they don't have to.
If they can confirm that you are violating their ToS first-hand and have the evidence to support it, then they don't have to wait for the system to identify and process the punishment. They can escalate the issue immediately.
If they state "troll and you won't have your account long", then that is probably the final warning you are being given that your behaviour is unacceptable.