There's a mental aspect (chalk = business time) but it's not a placebo. Have someone deadlift with a particular grip with and without chalk, and they'll lift more with the chalk, no question. It helps grip.

Can anything literally be only part placebo? obviously hyperbole, but anyone who has stuck their hand in a chalk bag knows how comforting it is in spite of it being the 7th time on that same rest hold.

I (as a somewhat mediocre climber) love chalk when climbing outside because it tells me what the next hold likely is :)

Agreed, dries hands and magically builds confidence. The ratio depends on weather and grade of the climb.

And the type of stone. There are places in Colorado where I don't bother with chalk because it's sandstone and is so gritty that chalk is ineffective.

On Chamonix-area granite, chalk is essential for me. Makes a huge difference. Definitely depends on the rock and the amount of sweat your hands produce.

When I was a math grad student, most of the other grad students hated getting chalk on their hands because it dried their skin out. Whereas I'd grab extra chalk from the board, and rub it over my hands to keep them dry.

Chalkboard chalk is calcium sulphate, whereas climbing chalk is magnesium carbonate.

You would not want to use chalkboard chalk for climbing, as it would make holds actually slipperier.

Some climbers do carry a piece of chalkboard chalk to mark key holds, though.