Chapter 17:

"I'm feeling thirsty," Harry said, "and that is not at all a good sign."

Dumbledore entirely failed to ask any questions about this cryptic statement.

He doesn’t seem to choke after this, but there follow several occasions where might have, had he been drinking. Anyway, the sentence means he kind of does use the Comed-Tea to kind-of-sort-of-predict the future, albeit not systematically.

Regarding the counting, his line in chapter 14 might be meant to suggest he had been doing more experiments “not on camera”. There are only three occasions where he’s seen using it until then; he shouldn’t have been that frustrated about the explanation after that few tries.