I'd say that people overestimate the role of the European parliament. It's basically a "pension fund" for politicians that are offloaded by their country's parties.
And since the UK Brexit nutjobs are leaving in less than half a year, the swing to the right, if one says there was one at all, will be somewhat neutralised.
Well, but the EU actually decided said copyright guideline (which the countries have to adapt into national law now) a few months ago. This was controversial, especially because of effects on the internet.