In my experience the red cups were used for mixed drinks being served by amateur bartenders (or jungle juice), or for kegs, or for drinking games. The latter was by far the most common - beer pong, slap/stack cup, etc. Often, people would rinse and reuse them to cut on waste and/or cost.
It has a lot to do with the 21 drinking age. Since getting alcohol is difficult without a fake ID or someone to buy it for you, you're dependent on what's at the party. Can't bring your own case of beer because you're 19 without a fake.
The host will get 1 or 2 kegs and make some jungle juice in a big cooler, again because they're probably under 21 so it's easier to buy only 2 types of alcohol
Have you considered that these are not SOLO (C) branded plastic cups? I don't think Top/Bottom/Switch was social media discourse prior to 2009 when SOLO (C) switched to their square cups.