Like Jannis said, you need to use a CMS, which will solve this problem.
But to give you more background info, here's a breakdown of the main areas of web development:
Static vs. Dynamic Sites
A static website is what you described in your question initially—where every page is hard-coded HTML and written by the designer. These sites were quite common in the 90s, and web developers/webmasters were hired to both write and maintain them. Back then, just having some rudimentary HTML knowledge qualified you to be a web developer, and most websites were coded this way as simply a collection of simple static webpages.
During this time, WYSIWYG (What You See Is What You Get) editors also came around. This allowed people who didn't know HTML to create webpages. Some editors like Dreamweaver also provided ways for web designers to specify an area of the page as editable, and clients could use their own copy of Dreamweaver to update the content on their site.
However, even fairly early on there were dynamic websites. These sites were more like applications, and their pages were dynamically generated when they were requested by a visitor. This meant that webpages could respond to the interaction of users, and content could also be personalized for each user. A search results page or a user dashboard are examples of dynamic webpages.
Server-side vs. Client-side Programming
On a static site, there is only client-side code, and the content is embedded in the HTML of each page. The pages can only be changed by manually changing the HTML.
Server-side programming is what makes dynamic webpages possible. This is code that is processed by the webserver instead of being sent to the browser. The server-side script run by the webserver would generate dynamic webpages that are then sent to the browser. This was first commonly implemented via CGI with server-side languages like Perl or C. But today other server-side languages like PHP, Ruby, ASP.NET, JSP, Python, etc. are also available.
Server-side languages allow you to write web applications instead of just static pages. The most common type of web application is a Content Management System that allows users to update their website without knowing HTML or editing files directly. This is the preferred way to manage a website today.
Dynamic websites require both server-side programming as well as client-side programming. Additionally, unlike static sites the data/content is usually separated from the HTML files and stored in a database, such as MySQL, PostgreSQL, SQL Server, etc. This has many advantages, including the ability to search, organize/index, and manipulate/edit the data more easily. This is usually done through a database language called SQL.
The HTML is usually stored as templates, reducing the amount of duplicate code/data, and making it easier to change the design of a website by making changes in a single place rather than in hundreds of different individual webpages.
Today, most sites, even personal blogs and websites of small mom & pop stores are dynamic sites with a CMS. This not only gives the user more control over their site instead of being reliant on a web developer, but it also makes it easier to maintain the site from a technical standpoint. The way static sites are created makes it very difficult and inefficient to maintain sites with more than a dozen pages. It also makes it more costly to make changes to the site's overall design.
So if you're a freelance web developer, you need to know a server-side language. Otherwise, you'll have to focus on front-end development and work for a web studio or partner up with a back-end developer. A freelance web developer who supplies static sites to their clients just won't cut it these days. It's unprofessional and actually hurts the client in the long run—especially as there are so many free and open source CMSes out there that most high school students could install and get up and running in just a few minutes.
If you're a web designer, then you could just install an open source CMS like Wordpress, Drupal, etc. for your client and design a custom template for them. That is probably the best deal for most business clients.