The EU has established an ambitious policy framework for demand-side mitigation in buildings towards net-zero targets. Here, we conduct a comprehensive quantitative assessment of 384 demand-side policy combinations for residential space heating that complement supply-side decarbonization efforts. We show that the implementation of EU Emissions Trading System 2, even when combined with deep decarbonization of energy supply, falls short of climate targets. Beyond ETS 2, we emphasize the need for ambitious subsidies for heat pumps as a critical component of a successful strategy. Conversely, a large-scale generic ‘Renovation Wave’ modestly contributes to decarbonization, is not a cost-effective strategy at the EU level and requires significant increases in public spending. We advocate for the implementation of a carbon tax, paired with substantial subsidies for heat pumps and targeted incentives for home insulation by country and building. This approach supports the decarbonization of the residential sector, limits the strain on the electricity grid, and alleviates energy poverty.