I think you are experiencing something that is pretty much a bug/issue in Joomla (editors):
You have to place the cursor in the editor for any of the editor buttons to work.

No, all works fine for me...

Using JCE here too, for testing.

Can you give me (super) admin access so I can take a look?
You can use the 'Confidential information' button in the forum editor to hide sensitive information.

You have disabled the editor button completely, it seems.
Content Templater needs that to function.

Ah, was expecting the editor buttons to be under the editor (which is default behaviour for JCE). I indeed see that having the editor buttons in the dropdown (Joomla icon button), then it doesn't work. Seems more like a JCE issue. But I'll try to look into this...

That's not what I am saying.
I meant that the button in the toolbar (top right) does not work when JCE is showing the editor buttons in the dropdown.
If you show the editor buttons under the editor as separate buttons, then the button in the toolbar does work.

Well, looks like the way JCE loads the editor buttons in the dropdown, causes things to function differently.

You can ask JCE to look into this. But I will look into this too, to see if I can somehow work around this issue. But not sure when I will be able to get that done...

So yeah, this looks like it is a JCE issue.

When editor buttons are rendered (via method onDisplay), they receive the editor name. For articles, this is "jform_articletext".
But when JCE renders the buttons in the dropdown, the value "__jce__" is passed to the editor buttons instead.

So when selecting a template from Content Templater, it is telling it to insrert it in the editor with the name "__jce__", which of course doesn't exist (as it should be "jform_articletext").

In other words: JCE is not passing on the correct editor name to the editor buttons.