I have two problems with that statement. First off, in practice very few people actually do vote based on their own selfish interests unless you're using a really broad and unusual definition of selfish. People largely vote as a form of signaling - they vote to send a message about how much they care about stuff and to have something in common with their group. (see also: _The Myth of the Rational Voter_)

The bigger problem is that there's a logical disconnect between your two clauses. The fact that people vote based on their own selfish interests does NOT actually guarantee the winner will serve their interests. It only would do so if people were really good judges of character and politicians were bad at lying and there were some reliable way to hold politicians to account for what they do.

Politicians lie about what they want to do, and we vote for them because we like the lies they tell. When everyone votes for the liar who promises to help everyone's own selfish interest, we get an unusually good liar who actually pursues his own interests when elected.

In the Bush v. Gore debate, Bush claimed to be firmly opposed to "nation-building". People who voted for Bush on that basis did NOT get a winner who was serving their interests.