I think the article summarizes pretty well what the drawbacks of POST are: unclear idempotency (well it's actually pretty damned clear: they are not cacheable). That complicates caching logic, and that's not just for the application server itself, but any reverse proxies in front of it as well as the user agent itself.

I'm not sure QUERY is a great solution, because in the context of a web application absolutely no one enjoys using a page that does not keep its state on refresh, so that really limits where QUERY makes sense, but if you have a case that is not driven by navigation, great.