I’ve been a big fan of the Post Kinds Plugin as well. Honestly I wished that WordPress had gone the extra mile and adopted something more like it when it was working on the Post Formats concept a few years back. I’ve written a bit about the Post Kinds Plugin in the past and perhaps you’ll appreciate some of those pieces, particularly the bookmarklet portions for desktop and details about mobile posting.
Because Post Kinds and Post Formats are not one-to-one or onto functions, doing the mapping in both directions is difficult, but when posting using a Post Kinds first method, you should be able to set the Post Formats you prefer. There are some useful defaults within the plugin, but they can be manually changed in the code available in the class-kind-taxonomy file in a relatively obvious way. In my case, while the mapping of “notes” to “asides” is a useful one, I prefer them to map to “status” for my current theme, so I just manually change that one word in the code to reflect my particular preference.
There’s a lot hiding under the hood if you want to tinker in the code. If you have issues or feature requests, I know that the developer David Shanske is very receptive to feedback towards improving the set up. (And similarly for almost all of the IndieWeb-related plugins which can be found on GitHub.)