Rather than thinking of the dash as being used for an “interruption,” it will serve you better to frame that rule as “A dash is used for a sentence that got started and did not get finished — broken sentence structure.” This thinking will allow you to use a dash in some places that may feel uncomfortable but where it is oh so correct. Sorry.
…The black car — was it in the right lane?
…The keys that I am holding in my hand — do these look familiar?
…Ms. Mary Reynolds — was she being considered for the job?
These are sentences/thoughts that got started but didn’t get finished in the original form, necessitating a dash.