Arguing to replace a well-defined single idiom (verbs) with some arbitrary combination of URI, custom headers, request body contents is exactly the opposite of what you want. Using defined verbs instead of having each site do their own slightly different thing makes APIs easier to discover / consume.
Also, I feel like "Execution in the Kingdom of Nouns" is semi-relevant here: http://steve-yegge.blogspot.com/2006/03/execution-in-kingdom...