Losing customization capability with managed servers (AWS or Azure)?
For platform-as-a-service (PaaS) databases, you don't get to design the underlying hardware such as CPU, I/O or memory configuration.

You can only choose a service tier that gives you a general idea of the overall performance. On Azure SQL Database, the different service tiers are defined by something Microsoft calls "DTUs", a performance metric that blends disk and CPU performance, but you cannot prioritize one over the other. I suspect AWS RDS works similarly in this regard.

The reason (personal opinion) why you would choose PaaS over IaaS (installing SQL Server on a VM) is that you don't have to maintain and patch any infrastructure. Your colleague is correct in saying that this comes at a price: less freedom.

The upside with PaaS is that you can scale it up and down whenever you need, minute by minute or hour by hour.