The Cloud (Azure, AWS, etc) are great solutions and products. In some scenarios. In others, they don't make sense.

You can scale up and down, and pausing things certainly can work. however you need to be able to pause them, and restart them. That isn't always feasible. I know some people with fairly steady workloads. In that case, it's not even necessarily cost effective to move to the cloud.

As David mentioned, your apps need to work in the cloud, and some of them will work well, some may not. Certainly some may need rewriting, but that's a separate issue.

Do you worry about hardware? Yes and no. You don't have responsibility, but choosing levels of hardware, weighing the costs of those, managing multiple connections to storage, or even managing the federation/sharding of data is your concern, and that can be affected by the performance of the underlying hardware. So while you are abstracted in the cloud, ultimately performance matters and whether I'm sizing the machines I need, perhaps moving data among nodes, or choosing EMC/HP hardware, the effect is the same.

You do need to be aware of how the hardware choices (direct or indirect) you make can affect performance. You will still need to stress test to understand how (and when) you will scale up and down.