It’s all very well not “wanting to use” < and > with timestamps, but those operators can be converted into index scans, and a string-match… well, it can, but EWWWW.
Well, the error is occurring because you need to explicitly convert the timestamp to a string before using a string operation on it, e.g.:
date_checker::text LIKE '2011-01-%'
and I suppose you could then create an index on (date_checker::text) and that expression would become an index scan but…. EWWWW.