They buy a lot of space, and they work with lots of providers, but they don't own very many sites. So I guess they don't "have a datacenter" for the purposes of this thread. No one I know in the business thinks this way in the year 2020.
Designing and operating datacenter facilities is specialized work, and it's about compliance, auditing, risk management, electrical, plumbing, hvac and other skilled trades. The datacenter industry actually has very little to do with computers, so there is a natural split between the facility and the server / network equipment it houses.
Basically all commercial datacenter providers operate as REITs, which is tax advantageous but extremely limiting in some ways. Amazon can benefit from this (with lower pricing) without dealing with it themselves.
Owning can offer some advantages, but it also means you're with that site for the long, long haul. Efficiencies of designs are always increasing, so operating in an old facility costs you money. If you built the site to your own spec, good luck exiting -- the next owner will have to do a total overhaul to get it to industry spec and get customers.
Even if you have a 10 year lease, there are always ways to get out if you want to. Especially if you're Amazon.