Neocities sites are static HTML, meaning once your files are uploaded,they don't change. Neocities doesn't do POST, there is no database backend or code running on the server.
Now in 2017, CORS allows client side apps to do just about anything traditional web apps have done, without page tranistions, given an appropriate backend. The difference is that html rendered by javascript on the client typically isn't indexed by search engines, while html served from a domain are. That's one significant difference, there are others.
There's pre-CORS workarounds. You could build an HTML multipart/form-data POST form inside an iframe and submit it with JavaScript for example. And if the server API supports JSONP, you can inject a script tag into your page that loads the target content as a script which then executes the data in a callback to read it back into the client page.
If you don't care about REST, and you only need to do GET/POST requests with a limited set of mime types, you could make a client-side application that gets around the same-origin policy without using CORS. But it's ugly.
With CORS though, you can potentially make any type of request across origins, because the CORS preflight OPTION requests allow a server to specify what request methods and what headers a client from a certain origin can access at a given URI. And these days, you can find CORS-enabled third-party REST APIs that you can use with things like client-side OAuth to provide all the same functionality that you would have in a traditional web applications.
I've been thinking about using a completely split API/client-side app architecture in my own projects lately, because scaling a bare API is much easier and cheaper than scaling a web application.
So yes, CORS is just a security measure, but it gives you more functionality in a client-side application by allowing you to bypass an older security measure, the same-origin policy, in a very clean and explicit way.
There are third party services that provide form processing of this nature, and even complete database backends and user management APIs (I built one).