This is really interesting. Is it something to do with people wanting to feel as if they’re on the winning team, even though you can effectively declare you’re voting for someone and there’s still no way for anyone to prove it*?

Yes, pretty much this. Deciding who to vote for can be a pretty difficult and complex decision. People end up relying on "short cuts", so something like the bandwagon effect happens. There's even evidence that shark attacks can influence how people vote.