You are spot on. This false statement comes from the same misconception that started with "thanks the Cloud(tm)(r)(c) hosting my multimillion page views website is now free" that marketers spinned. There has been a continuous price drop in commodities, it does not mean that the cost of operating these commodities drop at the same rate.

A whole operations team is needed to attend the infrastructure, whether or not your company actually owns its infrastructure.

Granted, thanks to automation, an operation team can attend to a larger number of servers than before, but this cancels out with the growth in the number of servers needed as the application complexity grows.