riot employee has stated it is against the rules to ask other players to report someone
True, but simply asking for reports is considered pretty low on the negativity scale (you aren't going to be punished just for that). However, if you are throwing around accusations or using the threat of a report to harass or coerce them, that is going to be considered more severe.
regardless of circumstances
Threatening people with reports doesn't help anyone. More often than not you're just throwing more fuel on the fire. Instead of feeding into the negativity, you should just ask them to stop and otherwise not engage them.
It's very important to note that number of reports in a game doesn't matter. At all. Any number of reports will always trigger a review for that player's behavior, so asking for more reports is harmful (due to the added negativity as mentioned before) and there are zero positive benefits. This is why you should simply avoid doing this.
so im going to be reporting every single individual i see doing this. linking the above imgur in the report chat box
I get that you're upset over your punishment, but trying to get other people punished isn't going to help you. You should be trying to actually understand what you were punished for, which is not limited to what you are claiming it was. If you'd like to discuss your punishment here I recommend sharing your entire chat logs with us to review.
when i dont see "your report has done something" messages frequently
You're not going to. As I already said, simply asking for reports is alone not very negative. You won't get anyone punished unless they are regularly negative or are doing a lot more than asking for reports, the same way that you got punished.