"Service" is doing a lot of work in that sentence. As is the narrow band of time and games you're referring to that required online activations at all. For a long chunk of PC gaming even games with a CD key only performed an offline algorithmic check on the key.
So yeah, if you breach the EULA in a service you get banned. If you own a thing you don't get it destroyed for a separate infraction, though. Which is my point.
And honestly, I seriously doubt that they wouldn't get away with an account ban in the EU. There are many ways in which the EU does good stuff to curb abuses of so-called "services", including dragging Steam kicking and screaming into having a semi-functional return policy, but EULA-based infractions driving account bans hasn't historically been one of their pet peeves, and there are absolutely examples of people losing access to large libraries out there.