A blockchain would work but is not necessary, and such a scheme would just pushes the storage and computational burden of the bookkeeping onto other users of the same chain. It still wouldn't help you if no provider were willing to honor your entitlement, which is what's going on here. You could achieve the same effect, and have a very similar set of failure modes, with digital signatures and emailing product keys or QR codes or whatnot into people's emails.

A few months ago, news broke that UltraViolet [1] would shut down. It was a meta-DRM scheme that, for all practical purposes, was equivalent to a mapping of 'Email Address' -> 'Set of Entitlements'. Various motion picture rightsholders and digital media-serving providers integrated with it, so your rights could be imported and exported into their central list and grow over time. Although the service cost money to run, its decommissioning and the aftermath of your entitlements is a question of political will, and not a question of any technical reality. Vudu is one of the very few providers through whom you can bridge your rights over to its competitor-turned-replacement, Disney's Movies Anywhere.

If there were the political will to do so, the damn hashmap of rights could've transferred directly, but it didn't. Altering the storage scheme would've altered the basic economics of UltraViolet's hosting, but it's not an armor against studio after studio withdrawing from issuing and honoring such rights, as it actually happened.

[1] https://en.wikipedia.org/wiki/UltraViolet_(system)