The original answer to "why does FastMail use their own hardware" is that when I started the company in 1999 there weren't many options. I actually originally used a single bare metal server at Rackspace, which at that time was a small scrappy startup. IIRC it cost $70/month. There weren't really practical VPS or SaaS alternatives back then for what I needed. Rob (the author of the linked article) joined a few months…
Flexibility.
When Netflix wanted to start operating in Europe, we didn't have to negotiate datacenter space, order a bunch of servers, wait for racking and stacking, and all those other things. We just made an API call and had an entire stack built in Europe.
Same thing we we expanded to Asia.
It also saved us a ton of money, because our workload was about 3x peak to trough each day. We would scale up for peak, and scale down for trough.
We used on-prem for the parts where that made sense -- serving the actual video bits. Those were done on custom servers with a very stripped down FreeBSD optimized just for serving video (so optimized that we still used Akamai for images). But the part of the business that needed flexibility (control plane and interface) were all in AWS.
Why would a startup use the cloud? Both flexibility and ease. There aren't a lot of experts around that can configure a linux box from scratch anymore. And even if you can, you can't go from coded-up idea to production in five minutes like you can with the cloud. It would take you at least a few hours to set up the bare metal the first time.