Economy of serverless is the biggest benefit of this architecture. There are other benefits, but most of them are not new. We had database triggers in the past. But they are now easier then ever, fully managed and available to everyone.
Serverless is about the cost. But not just infrastructure costs, which can be low. For example, Vacation Tracker costs $0 for more than 150 teams using the app. It’s also about development costs, maintenance costs, opportunity costs, etc.
On serverless vs PaaS: I would say that serverless is an evolution of PaaS and BaaS.
On serverless beyond functions: I don’t know about Cosmos DB, but AWS DynamoDB has auto scaling, pay-per-use and requires minimal maintenance. For me that’s enough to be considered as serverless. Name might not be the best one ever, but the name itself is not important.