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CO2 emission reduction of sectors such as aviation, maritime shipping, road haulage, and chemical production is challenging but necessary. Although these sectors will most likely continue to rely on carbonaceous energy carriers, they are expected to gradually shift away from fossil fuels. In order to do so, the prominent option is to utilize alternative carbon sources—like biomass and CO2 originating from carbon capture—for the production of non-fossil carbonaceous vectors (biofuels and e-fuels). However, the limited availability of biomass and the varying nature of other carbon sources necessitate a comprehensive evaluation of trade-offs between potential carbon uses and existing sources. Then, it is primordial to understand the origin of carbon used in sustainable aviation fuel (SAF) to understand the implications of defossilizing aviation for the energy system. Moreover, the production of SAF implies deep changes to the energy system that are quantified in this work. This study utilizes the linear programming cost optimization tool EnergyScope TD to analyze the holistic French energy system, encompassing transport, industry, electricity, and heat sectors while ensuring net greenhouse gas neutrality. A novel method to model and quantify carbon flows within the system is introduced, enabling a comprehensive assessment of greenhouse gas neutrality. This study highlights the significance of fulfilling clean energy requirements and implementing carbon dioxide removal measures as crucial steps toward achieving climate neutrality. Indeed, to reach climate neutrality, a production of 1,046 TWh of electricity by non-fossil sources is needed. Furthermore, the findings underscore the critical role of efficient carbon and energy valorization from biomass, providing evidence that producing fuels by combining biomass and hydrogen is optimal. The study also offers valuable insights into the future cost and impact of SAF production for air travel originating from France. That is, the European law ReFuelEU would increase the price of plane tickets by +33% and would require 126 TWh of hydrogen and 50 TWh of biomass to produce the necessary 91 TWh of jet fuel. Finally, the implications of the assumption behind the production of SAF are discussed.