Political advertising on social media has fundamentally reshaped democratic deliberation, playing a central role in electoral campaigns and propaganda. However, its systemic impact remains largely theoretical or unexplored, raising critical concerns about institutional fairness and algorithmic transparency. This paper provides the first data-driven analysis of the relationship between direct democracy and political advertising on social media, leveraging a novel dataset of 40,000 political ads published on Meta in Switzerland between 2021 and 2025. Switzerland's system of direct democracy, characterized by frequent referenda, provides an ideal context for examining this relationship beyond standard electoral cycles. The results reveal the sheer scale of digital campaigning, with 560 million impressions targeting 5.6 million voters, and suggest that greater exposure to "pro-Yes" advertising significantly correlates with referendum approval outcomes. Demographic microtargeting analysis suggests partisan strategies: Centrist and right-wing parties predominantly target older men, whereas left-wing parties focus on young women. Regarding textual content, a clear pattern of "talking past each other" is identified; in line with the issue ownership theory, parties avoid debating shared issues, preferring to promote exclusively owned topics. Furthermore, the parties' strategies are so distinctive that a machine learning model trained only on audience and topic features can accurately predict the author of an advertisement. This article highlights how demographic microtargeting, issue divergence, and tailored messages could undermine democratic deliberation, exposing a paradox: Referenda are designed to be the ultimate expression of the popular will, yet they are highly susceptible to invisible algorithmic persuasion.