In a limited support scenario, probably it makes most sense to ask how society, administrations and governments in general, and the European Commission in particular, can benefit from open source software, and how they can use it for their own advantage, rather than the other way around (how can the open source movement benefit from governmental support).
In this case, they will probably invest some time and resources to assess the feasibility of open source software in their areas of interest, and to identify the barriers which could impede their adoption of open source technologies. With time, they will identify some strategic open source projects where they want to contribute, and will understand more clearly some of the benefits in terms of flexibility, usefulness in the entire life cycle, and adaptability. They may be interested in trying open source software for several (especially mission critical) components of their infrastructure.
In the long term, this will give results, especially in the form of greater acceptance of open source software in society in general, because of the amplifier effect that its use in governments has on society. If they finally recognize the benefits of the open source model, probably they will also help to overcome the future problems (specially those related to the legal framework) that have been described.
The recent case of the German government funding the development of GPG (GNU Privacy Guard) because it is found to be of great benefit for society shows how this scenario may develop in the short to medium term.