In light of climate change and security concerns, decarbonisation has become a priority for industrialised countries. In the European Union (EU), decarbonisation scenarios used to support decision-making predict a steady decrease in greenhouse gas (GHG) emissions, mostly driven by changes in production mixes and improvements in efficiency. In the EU's decarbonisation pathways, the power sector plays a large role, reaching zero emissions by 2050. From a biophysical perspective, decarbonisation becomes not just a matter of replacing carbon-intensive with carbon-neutral electricity flows, but also a matter of building and maintaining new infrastructure (funds) which, in turn, is associated with GHG emissions. By not accounting for the emissions associated with funds, particularly those required to increase grid flexibility, scenarios used to inform decarbonisation narratives in the EU are missing a key part of the picture. We show that a rapid and deep decarbonisation of the EU's power sector through a production-side transition between the years 2020 and 2050 leads to cumulative emissions of the order of 21-25 Gt of CO2 equivalent, within a range of approximately 35-45%. The results are obtained by modelling two decarbonisation pathways where grid flexibility increases either through storage or through curtailment. The analysis suggests that scenarios informing decarbonisation policies in the EU are optimistic and may lead to a narrow focus on sustainable production transformations. This minimises the perceived urgency of reducing overall energy consumption to stay within safe carbon budgets.