We may never see Half-Life 3 in our lifetime and it is because Valve wanted to work on a VR game to “show off” their game-making skills instead. The developer is ramping up for its release of Half-Life: Alyx, a VR game based in the Half-Life universe which takes place between the original game and Half-Life 2.
In an interview with IGN, designer Dario Casali revealed that over the years the company had tried to repeatedly continue the story where Episode Two left off. Valve planned on releasing an episode every year in order to get more content out faster. However, this did not work out as development time grew considerably longer as the company began working on episode 2 and 3. This resulted in them abandoning the project.
It was also coupled by the fact that Valve was, at the time, working on its engine Source 2. They claim there was not enough resources to keep both projects at full steam and did not want to develop a game for the sake of it. Instead, Valve claims they preferred to release a game that would be as revolutionary as the original game and sequel. The answer to this was the VR game Half-Life: Alyx as the developer wanted to use the technology to show off their game-making skills.
So in short, Valve wanted work on a game that would showcase their skills as a developer and releasing Half-Life 3 or Episode 3 would not do that. A VR game, on the other hand, would be a better showcase of their game-making skills.
So will we ever get a new game? No one knows right now but it is possible. Perhaps with the release of next-gen hardware, Valve will see it as an opportunity to showcase their game-making skills and release a sequel in the series.