It doesn't. There are plenty of bourgeois democracies that don't use FPTP for all their voting: Japan, Australia, South Korea for some of their elections. Doesn't make a difference (except it might make the bribery a bit more expensive, since you have to buy off more political parties than just two).

The fundamental problem is capital standing above political power. If it does so, then no amount of alternative voting systems can fix the issue. Socialism is the only answer.