Oh my god. After a year of hopelessness, Petr's comment was the answer. Thank you so much!
I mean, this has to be a bug.
In Helix Ultimate settings, 'Blog Image Sizes', i left the Medium, Small & Thumbnail options 'unchecked' and height was indeed 0. Now it turns out, that the height listed in (one of ) these unchecked options, is actually what has been preventing Blog Media from working.
I filled in dimensions for all image sizes, (still left only Large as checked, the rest unchecked) and now when I go create an article and upload an image via Blog Media, it works perfectly. Wow, just wow. Not fully logical in my opinion but a solution is a solution. :-)
I'd consider this forum thread resolved.