Not always - you can lease your servers from the vendor as well, in which case you're renting the rack space, power and cooling from the datacenter and you're renting the servers from the vendor - most of the leases are designed so you can refresh your hardware every 4-5 years and it's usually still cheaper than renting from a dedicated hosting company.

Once you have an established baseline for your server needs - it's almost always more capital friendly to buy the servers and keep them running for the ~5 reliable years you'll get out of them - usually break even here is 2-3 years vs renting from a provider. If you're running your servers until they fail you'll get 7-10 years out of them, provided the power cost is still worth running them (usually that is also around the 8-10 year mark depending on your power cost).

So there are many reasons you'd buy vs rent - including capital deductions and access to cheap interest rates. You can also get some pretty crazy deals (like 33% of new price) by buying 2-3 year old equipment, then continue to run them for another 4-5 years, which is the lowest cost scenario if you don't need bleeding edge.

What about the cost of having people actually go to the datacenter to install hardware, and go again whenever there is a hardware problem, possibly resulting in much longer downtimes than with a rented server?

Especially for the "one (or a few) big server" scenario in the article, that would seem to me a pretty big factor.