This is an idea that might not pass a cost:benefit test, but I'm offering it up for your consideration anyway.

Situation: You have a ref in an article that is used several times: <ref name="Foo">. It's a citation to a book, and someone has accidentally formatted it in {{cite web}} rather than {{cite book}}. You want to change the template, but keep the same contents.

Solution in wikitext editor: Find ref that actually contains the citation template, remove 'web' and type in 'book'. Save page.

Solution in VisualEditor without Citoid: Select any citation to it, choose Cite>Basic, select and remove the entire citation, and then Insert>Template and re-create the whole contents.

Wrong solution in VisualEditor with Citoid: Select any citation to it, remove the entire citation, and select Cite>Magic to make a new one. (This is "wrong" because this will only change the selected instance of the citation, and not any of the named copies.)

Adding an option to change the name of the template while keeping all the parameters and content would solve this problem (and all similar cases for all templates) but would probably result in a complicated experience for relatively low benefits. Adding an option to change only the citation template within a ref is a smaller solution that might be wanted anyway (what if Citoid says that my magazine article is a "journal" rather than "news"?).

However, it's possible that this one scenario could be solved this third way:

Select any citation to it, choose Cite>Basic, select and remove the entire citation, and then (from inside the basic reference dialog) select Cite>Magic to automagically re-create the contents, while retaining the original <ref name>.