According to the article:
This new legislation requires every Spanish publication to charge services like Google News for showing even the smallest snippet from their publications, whether they want to or not.
This sounds as though it won't matter if Spanish news aggregators move abroad, because the law applies to the Spanish news sources themselves, not the aggregator. Unless I'm misunderstanding (and maybe I am, so correct me), if a news publisher inside Spain doesn't demand payment from an aggregator located outside Spain, the publisher in Spain gets punished by the Spanish law.
This approach would be a nasty way to do as you said: limit the ability of smaller, alternative news sources, to attract readers. Unless they left Spain, they would not be allowed to volunteer their content to any free aggregator anywhere, and if they DID leave Spain, they would lose the ability to report on local Spanish issues.