A situational issue that I've run into a number of times with GET is various middle boxes inadvertently logging sensitive data that was in the query string of a GET. By sensitive data, I mean PII and CU information. While we have used POST to side step the issue, it'd be nice if there was a more appropriate method.
But a new HTTP method would require an update to the middle boxes... if you're doing that, you could also consider updating them to avoid the issue you described.
Overall, I'm not necessarily inherently against new HTTP methods. If equipment that is supposed to be transparent to HTTP isn't, should be solved elsewhere, not by a new HTTP method.
Many logging systems exclude query parameters from the logging by default. It is also fine to include PII data in the logs as long as they are handled correctly and deleted after 30 days. But I have got burned by this problem as well. Like people using PIIs for identifiers.
