FWIW, inserting Template:Cite interview with VE works for me on enwiki. My screen does not look anything like the OP's however, so there may be some interaction with the browser, user scripts, user preferences, or something else that does not match my configuration.

Oh, interesting. Adding and editing an existing one in my sandbox worked as expected. But when I put <ref> tags around it, it stopped being able to be edited correctly.

I've put examples up at https://en.wikipedia.org/w/index.php?title=User:Goldenshimmer/sandbox&oldid=960065161

{{u|Jonesey95}}, would you be willing to see if the two citation templates in that revision labeled "Not editable" show up correctly for you in visual editing? Thanks!

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Ah! I figured out a way to edit the ones that have that problem — click edit on the ref, then click the displayed ref, then click edit in the resulting pop-up, and it makes a second pop-up which works correctly.

So it does seem possible to edit them, it just has an extra, non-obvious step that the Web ones don't.

The single-step editing is only available for the few most common templates, listed here: https://en.wikipedia.org/wiki/MediaWiki:Visualeditor-cite-tool-definition.json.

Administrators can add more templates to that list using these instructions: https://www.mediawiki.org/wiki/VisualEditor/Citation_tool#Additional_types.

Doing that will also add an entry in the Cite → Manual menu.

I see, so it would need to clutter up the menu to be able to be detected then...

I guess this is a request to either separate that dependency so it can open the usual citation tool when there's only a single citation template inside the ref tags, or to otherwise clarify the user interface somehow.

I don't think this is common enough that it needs to go into the main menu, probably.

Showing a hint the first time it's opened to the effect of

"The contents of this ref tag consist of a single template. To edit the template, click the displayed template in the ref tag edit box and choose the Edit option."

would make this clearer. I didn't try clicking on it at all until I was trying to understand it to write up this report after the fact, since I thought it would overwrite the template with the rendered output if I edited the displayed text.