Well you can't reliably use GET with bodies. There is the proposed SEARCH but using custom methods also might not work everywhere.

No, QUERY. https://datatracker.ietf.org/doc/html/draft-ietf-httpbis-saf...

SEARCH is from RFC 5323 (WebDAV).

The SEARCH verb draft was superseded by the QUERY verb draft last I checked. QUERY is somewhat more adopted, though it's still very new.