Proxies were around before NAT, and so early on in the internet, they were popular ways to get an entire network online with only a single public IP. NAT initially required complex software and/or hardware to get it working, and so understandably some organizations still used proxy servers to provide access.
In the last decade however, NAT has become immensely popular, and is available in virtually every router on the market. The hardware is so fast and cheap that unless you are trying to use a $50 home router to run say, a 5000 person office, you're not going to have issues. Firewalls are also smart enough to do complex authentication, easily allowing different users to access different services at different times of day (even the most basic home routers often have this functionality built-in). In fact, NAT is likely easier to set up and manage than a proxy server.
In my job I still often deal with clients using proxy servers to provide internet access, and the ensuing issues that arise from authentication, what account services are running as, etc. This includes some companies that are very big and old (and so I guess got their network installed over a decade ago, and just kept it that way) as well as relatively new organizations that have only a few years old.
So my question is, why in the heck are there still so many places using proxy servers for this purpose?