"Streaming subtitled box sets" are not "the new Eurovision", and Eurovision itself meant and means squat when it comes to "common European culture". (If anything Eurovision is something most of us watch for laughs, and more often than not we take sides and vote for countries we already knew and liked for other cultural and historical reasons).
We have had and have a quite good understanding of other European states and their cultures, even more so when they share the same borders (as it's often the case in Europe - to have 2, 3 or more countries sharing your borders and have millenia of interaction with them and their culture, plus "shared" areas of mixed culture, e.g. Strasbourg).
It's not the lack of Netflix's European programming or watching each other's shows that prevented a "common European culture".
Not to mention, we have had something like that, both in the high-brow sphere (for centuries), but also in the pop-culture.
E.g. non-artsy Italian and French cinema from Alain Delon to Bud Spenser was big all-over Europe until the 80s when Hollywood dominated with emphasis on impossible to meet big productions and vfx. European pop music frequently travelled all over Europe, from Al Bano to Roussos and from Nina Hagen to Kraftwerk. Comics too, French, English, Italian, Belgian, Spanish (to namedrop some that were published and loved all over Europe: Asterix, Tin Tin, Mortadelo y Filemon, Lucky Luke, Blek, Judge Dredd, the Smurfs, Manara, Corto Maltese, and so on).
And as somebody mentioned, Interrail and Erasmus have been going strong for 3 to 4 decades now or more... (I've done the first, and have had many friends doing both the first and the second.).