Got you and I can reproduce the same behaviour.
I think what you’re seeing is a file path/permissions issue rather than a syntax problem as all the apps mentioned support the standard Markdown image syntax.
If you try it with a web link for the image rather than a local file then that should work anywhere.
I found the following in the iA Writer documentation…
When using local images, they must be in a folder added as a Library location. This gives iA Writer permission to use the file. Folders can be added to the Library in Preferences on Mac.
The easiest way I found to do this is to get the iA Writer and Finder windows up side by side and drag the 1 Writer folder from the Finder into the “Locations” section of the sidebar in iA Writer (it doesn’t actually move the folder just creates a reference to it in the iA Writer library).
If you do this then the images from the 1 Writer file display correctly in the iA Writer preview.
Appreciate you may prefer a different tool for other reasons, but just wanted to make sure you weren’t rejecting iA Writer for lacking something that it can actually do.