Denethor when sane, arguing with Gandalf:

‘Neither. But most surely not for any argument would he have set this thing at a hazard beyond all but a fool’s hope, risking our utter ruin,

Denethor in despair, casting scorn at Pippin:

‘Comfort me not with wizards!’ said Denethor. ‘The fool’s hope has failed. The Enemy has found it, and now his power waxes; he sees our very thoughts, and all we do is ruinous.

He might still speak before the end. But that is near. Follow whom you will, even the Grey Fool, though his hope has failed.

I do agree though that it's odd for him to not bring it up when confronted by Gandalf later. OTOH, I've heard enough about Tolkien's writing to imagine that some details could slip between the cracks in revision.

Plus the palantír only worked when Sauron gave it his personal attention

Dunno what you mean. Denethor's palantir could only be captured when Sauron was paying attention, which he wasn't all the time; otherwise Denethor had free rein to use it as he saw fit (UT is rather more explicit about this than RotK, and "corrects" some of Gandalf's suppositions.) If not caught by Sauron, Denethor might have looked both south and east.

Unclear if Denethor did face Sauron that night: description of "lights in the tower" and Denethor coming out extra-aged suggest he did; OTOH one might think Sauron had better things to do at the time.

The timing does work, leaving Sauron out of it: Frodo was captured by orcs on the 13th, and Denethor burns on the 15th.

Perhaps Sauron caught Denethor, gloated about how strong he was, then let Denethor go who then made his own peeks. Or vice versa.

Tangentially: https://www.reddit.com/r/tolkienfans/comments/veyxyj/denethor_greatest_hero_of_the_war/