In the recent IGN interview, Dario Casali talked about three key points why we never saw Half-Life 2: Episode 3.
Dario is in Valve since 1996, and he's one of the key people in the company with the knowledge of all internal processes. When asked about Episode 3, he listed three main reasons as an explanation of why it never happened.
As one of the main reasons, he blames "scope creep." He said that Episode 1 was produced in a ~year, while Episode 2 already took two years to be finished. With Episode 3, it looked like they are working on a full-blown sequel and not a small "episode" type of a game they initially wanted to do.
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"Scope creep" is a known problem in video game development, when the game starts to grove exponentially, being much bigger than the original idea of it. Scope creep naturally leads to more and more work to be done in order to develop and implement all the new features, mechanics, story elements, and other additional stuff. Score creep is a problem mostly in companies with bad producers/managers since it's their job to oversee these things and make sure something like that will not happen. And Valve is known as a "free for all" working place, without a clear corporate structure, where everyone has the freedom to work on their own stuff and ideas. That sometimes might be great for creativity, but often it can lead to chaos.
A second issue, according to Casali, was the work on Source 2. They started developing it around the same time, and he says that they didn't want to "make the same mistake" as with Half-Life 2. They wanted first to develop Source 2 fully and only then to start working on a new Half-Life game.
Finally, he says that they don't want to see new Half-Life games as just another game. They want to push the boundaries and go beyond known standards with new Half-Life titles. He confirmed that they had several Half-Life games in development during the last decade, which were eventually discarded as they didn't feel groundbreaking enough.
Fans of the Half-Life series are not happy with these answers and think that they are all just excuses, judging by the comments on the Reddit thread.
You can read the full story with more details in this IGN article.