When Half-Life 2 came out it not only lived up to the legacy created by its predecessor, but created a league of its own. Everything about the game was perfect, and the series had thus revolutionized the FPS genre twice. Half-Life 3 is definitely in the works and there is good reason why Valve is so silent about it. In order to live up to the hype and expectations, Half Life 3 needs to be extraordinary and it has to be a powerful amalgamation of art and technology. “These things, they take time”.
Half-Life and Portal writer Erik Wolpaw has detailed how his team has been using Generative AI for research and development at Valve.
Valve has responded to New York Attorney General Letitia James' lawsuit against the firm, stating it does not believe that lootboxes in its Counter-Strike 2, Dota 2, and Team Fortress 2 games unlawfully encourage minors to gamble, adding it was "disappointed" to see the action go ahead despite efforts to "educate" the NYAG about virtual items since "early 2023."
Because valve wants it to be a WiiU exclusive title and doesn't want to piss off there fanbase too soon. So there keeping it under wraps for now duhhhh.
One of the main reasons HL3 has not appeared is because Valve has a completely (or almost) flat organisational structure.
People can work on anything they want. This things that attract attention and support of others get made. Gabe may be the de facto top dog, but generally anything goes.
So either, there's not enough appetite / support to make HL3, or Valve are waiting for the tech to be where they need it, or Gabe has vetoed it for now for some other reason. Or a combination of those factors.
Valve are not short of money, so it isn't as if they have to make HL3. Also, the game is predominantly associated with PC, so there's isn't the same incentive as for MS to have a Halo game for XB1 or Sony to have a Naughty Dog game - those games sell hardware units.
Valve does not YET have a reason to sell hardware - but Steamboxes and SteamOS should tell you all you need to know about when HL3 is likely to start being seriously developed.