They have mentioned before that they gave up on episodic development, which tacitly ditches Episode 3. The episodes ended up not being that much easier or faster to make and in a time when PC games in retail was still kind of relevant, it was a pain to make, distribute, and get shelf space for.

from Episode One and Episode Two, along with ideas and experiments for the third episode that never came to be.

Emphasis mine. They mean HL2 Ep 3 was planned and canceled. Not HL3.

They’ve admitted to cancelling ideas before, getting to various stages of production before going back to the drawing board, but always (and appear to still) insist that it is in development on some level. That’s why Newell’s responses to questions about hl3 are usually some form of “we have nothing new to share.” Valve doesn’t like sharing until they’re in the final stages of development, and hl3 has never made it that far.

Only those ideas and experiments never came to be. There’s still hope

No. They publicly stated the cancellation of episode 3 years ago when it was still relevant, and is one of the reasons I was always confused about every single rumor that HL3 was going to be a thing; they never announced a HL3 and they already said Episode 3 wasn’t coming out. Where the heck did the HL3 rumors come from?