In the past people still would rent more than they owned. They just traded having to visit a store with a selection limited by physical space for an instantly available digital service.
But you'd never show up to a Hollywood Video looking to rent Terminator 2 and get told that it's a Blockbuster exclusive rental. There were Blockbuster exclusive movies, but they all sucked so it didn't matter.
Netflix Streaming gave bigger selection and instant access. You could get 15 minutes into a movie, realize it sucks, and you didn't have to drive back to the video store for a different one.
Nowadays the selection on any streaming service feels like 1/3 of what a VHS rental place would have. Sure there's more actual quantity, but it's like the $1 rental bin. Currently, if you pick X popular movie and Y streaming service, odds are you cannot watch it on that streaming service.