> giving your biometric info away
...you do realize that "biometric info" is the most easily accessible / stealable type of information. Anyone can grab your biometric infos easily if you become a person of interest, so it's hard to think of what you're practically gaining.
It's like Americans with their SSNs, this kind of information is practically public (ignoring legalese and crap), so trying to keep it away from a big database will only work up to a point. In a way things like the Equifax hack are a good thing because now anyone's SSN can be assumed public.
Sure, if you fancy a career in intelligence or as an undercover something, it would be a valuable personal asset to have "biometrics not recorded in databases X, Y and Z". But it only works until you're caught/recorded the first time. And if you're either "interesting enough" or a public person, anyone caring to track you surely already has your biometric infos.
(Otoh, assuming anyone managing that big EU db will be grossly incompetent is probably realistic :P)