> Is an api that [...] uses only GET requests for [...] state changes really a horrible api?
Yes, it is. Because HTTP clients (including browser) rely on the common method properties as defined by the HTTP standard (GET having no side effects, PUT being idempotent, etc.). The simplest example is caching / proxying, but there is other behaviour.
I strong recommend acutally taking a look at the HTTP standard. The RFCs such as RFC 7231 are well organized and human readable. For the definition of the HTTP verbs, see:
RFC 7231 Hypertext Transfer Protocol (HTTP/1.1): Semantics and Content
4.2 Common Method Properties