Why does everyone random vote at the start of a game?

[Cogito Ergo Sum]

Indeed. Day 1 tends to be a problem, because there is usually hardly any info. Random voting is a means to try and get the game started for real. It is an attempt to make things happen, to give people something to talk about. It doesn't always work, but it can work.

If one person says they're not going to put a random vote, that can actually be helpful, as it can lead to discussion. If nobody random voted however, the game could stall quite quickly.