Plain static sites such as the ones you created with Netscape Composer will still work just fine. There is no technical barrier to putting one up -- just edit together some HTML files, upload, and voila.
The "site engines" you refer to are likely "content management systems" such as Drupal and similar. They are increasingly being used even for small sites because they can greatly simplify site maintenance. They offer in-browser editing of a page's content without any particular need to know HTML; the CMS offers WYSIWYG controls very much like the ones that you use here on this site (Bold/Italic buttons, live previews, and so on).
They also offer a wide range of expanded abilities, such as automatic creation of RSS feeds to syndicate your site's content, specialized modules for managing the workflow that goes into posting new pages, image management, and so on.
The tradeoff comes in the form of increased complexity. A CMS will typically require a a database, a correctly configured scripting environment (PHP being most common), and creating a series of "template" files defining the site's look and feel. Creating those template files (or "theming" the site) varies in difficulty depending on the CMS, but it always assumes that you have at least a basic understanding of the scripting language the CMS is written in, and also how the theme system for that CMS works. So -- the initial setup is more complicated compared to just uploading a bunch of static files.
The deciding factors are, basically, the size of the site and the technical proficiency of its maintainers. If it's a small site maintained by people who are comfortable editing the pages directly, then a CMS may not be worth it. But if it's a large site with a group of non-technical people who need to maintain it over time, then a CMS is well worth the trouble.