Virtually every company is moving toward this model - Microsoft, Adobe, Disney, etc. That's my main complaint.
Liberal economists use this term the most, they usually mean large companies trying to get goodies from the state. Lefty ones tend to mean something like "profiteering."
X-as-a-service businesses models aren't exactly new in intellectual property. The patent/copyright systems explicitly assume licensing and royalties. But, I agree that subscriptions are pretty new in consumer-land.
But we need a different term rent seeking already has competing usages.
I think here you might be thinking of Regulatory Capture.
No, they don’t. That is neither a correct nor common use of the term.