I'm a fan of multilingualism and think it should be defended at every level of government! The current 24 official languages of the EU should be closer to 50.
Mutual intelligibility already go a long way and it wouldn't be any different. Czech and Slovak people can work and study in each other's countries without ever passing any kind of language test. People in Scandinavia can choose to do university exams in their own language! Probably the same would happen with Catalan and Occitan, or Sardinian and Corsican, were they not neglected!
There's also translation and interpretation. We have the means and the money to have a completely functioning multilingual apparatus.
Multilingualism is good for society, cognition and culture
In cases where we need one language I think we could take a page out of Switzerland's book, where they use Latin for those cases (coins, namely). The EU already does with their website. I think another cool option could be Modern Indo-European, Lidepla or maybe using Interlingua, Interslavic and creating some sort of Intergermanic?
The problem with using one of those is you always leave someone out: everyone non-Romance with Latin, everyone non-IE with Modern Indo-European. Even with the last option you'd leave Uralic, Basque and Maltese out. Maybe the EU could fund a conlanging project to design a Intereuropean language for use in logos and names of institutions only? Because in general communication, as I already said, everyone should speak their own lamguage