If when you are attempting to add blocks to a Query Template in the site editor, they are not addable either by drag drop or via inserting and/or the inner blocks are no longer editable in the inspector, then this post is for you.
Note the toolbar block inserter disappears for the Query Template block:
Also note, the inner blocks are no longer editable in the inspector:
This is a known issue and happens if your query is not returning results.
Once your query is successfully returning one or more results you will be able to edit all of the blocks again.
Hopefully this saves you from a few moments of frustration.