> Even if some people don't vote, wouldn't the interests of the nation be properly (statistically) represented by the people who do get out and vote?

No. Harvard has (free!) [Five Studies on the Causes and Consequences of Voter Turnout][1]. The first one is a study of Australia before and after compulsory voting was implemented. The election results were significantly different (in terms of overall seats won by different parties) before and after compulsory voting, indicating that the original voting population was not representative of the whole population.

They go on to do pretty much what it says in the title--examine the causes and consequences of voter turnout, including examining how effective different get-out-the-vote campaigns are.

  [1]: https://dash.harvard.edu/handle/1/11156810