I don't see what Schengen had to do with anything - lack of passport checks is a nice convenience but that is all. Freedom of movement, sure, but that is separate from Schengen.
And I think dismissing light entertainment as not being culture is pointlessly elitist. A whole lot of culture is light entertainment, and often what spreads to the most people.
Most of Western Europe has close ties going back much further, but close ties does not translate immediately to the extent of shared culture diffusing through populations.
Not least because there is a stark difference between ties that are confined to a subset of the populations vs. those that are more widely shared.
I live in the UK, and have done so for 21 years, but come from Norway, and one of the first things I realised on moving here was how superficial cultural exchange often is - most people have no conception what it's like to live in another country. Even one they in many cases have often been to as a visitor.
And e.g. shared entertainment is part of that. Growing up watching different things, talking about different things etc. contributes to cultural differences even decades later. I still experience people bringing up cultural references that they share that I don't for that reason.
I agree with you regarding Erasmus for that reason, but I also think that seeing more shared programming will matter more than you think in giving more people shared reference points because it is a lasting and ongoing part of people's lives.
Irrespective of Netflix' motivations.