It seems that "slate" meaning "criticise" might come from the idea of
throwing a slate at someone.
slate, v.2
Etymology: apparently < slate n.1 Sense 2 appears to have
originated in Ireland.
slang and colloq.
2.
a. To beat or thrash severely.
b. Mil. To punish (an enemy) severely.
3.
a. To assail with reproof or abuse; to rate or reprimand; to scold
severely.
b. To criticize (a book or author) severely; to castigate, cut up.
slate, n.1
1.
a. A thin, usually rectangular, piece of certain varieties of stone
which split readily into laminæ ..., used especially for the
purpose of covering the roofs of buildings.
However:
slate, n.3
Etymology: < slate v.3
A severe criticism; a slating.
1887 A. Lang Bks. & Bookmen 19 ‘Slate’ is a professional term
for a severe criticism.
slate, v.3
Etymology: < Old Norse *sleita, corresponding to Old English sl?tan
: see sleat v.1
north. and Sc.
1. trans. To incite or set on (a dog). Also const. on, at, against
(a person, etc.).
2. To bait, assail, or drive, with dogs. Also fig.
--
Peter Duncanson, UK
(in alt.usage.english)