> I do wonder why FAANG aren't more active lobbying the EU, though

They are quite active. I.e. Google is said to have spent 30+ millions on lobbying about the copyright reform. Facebook is lobbying too: https://www.politico.eu/article/inside-story-facebook-fight-... (although not necessarily as one would think from their public statements, it seems like they're quite happy to tout how effective upload filters are when it seems like they can use that as an argument to avoid other rules)