AWS accounts still take management and a team of people that need to maintain a whole lot of different aspects of it, so you're not really saving on headcount. You're just moving that capex to opex.
It's important to be flexible enough to be able to deploy onto a cloud provider if the situation demands (e.g., new client demands infrastructure run in $FOREIGN_REGION_X where you don't already have a DC), but everyone's insistence on going 1000% AWS is absurd and IMO totally unjustifiable.
For what AWS provides, the DIY approach would be insanely expensive and wasteful. Not to mention, it would take years to build a basic MVP. They'd have to scope out the project, hire people just to design it, and then pay for several contractors just to stand up the first iteration of a working system (which will still not compare to AWS in terms of resiliency, redundancy, and accessibility), and then maintain it....forever. Also, many people already know how to interact with AWS. NASA would also need to design and maintain user-access methods to the data, on top of just plugging in thousands of hard drives and making them all work nicely together.
Why reinvent the wheel when there is a perfectly good wheel manufacturer that has already proven extremely successful at what they do?