The notification on the medication instructions refer strictly to adrenal insufficiency, that is, Addisons.
I don't know that it is routine in any medical setting that I have encountered to inquire about one's adrenal status. My first dr put me on levothyroxine without any of my medical records, aside from the TSH test indicating I was hypo and never questioned me. Adrenals were not addressed by any other the other practitioners I saw.
I presume drs are looking for classical Addisons symptoms that may be evident upon physical examination. So, I also presume, if they don't see them, they are not going to pursue it further.
In any event, as you will learn from reading here, adrenal fatigue/exhaustion, can interfere with thyroid hormone treatment. And many people here have sought to investigate this avenue through the, best way apparently, 24 hour saliva test. In the event that there is adrenal fatigue, many practitioners who recognize this as a legitimate medical concern, will put patients on cortef.
However, adrenal fatigue is not widely accepted in mainstream medicine. So, if you talk to your dr, and if he/she is mainstream, and does not subscribe to the belief that adrenal fatigue is a legitimate medical concern, you may expect to get the appropriate response, that there is no concern with taking the thyroxine unless you have clinical addisons.
That is not to say that you should not at least bring it up with your dr to see what he/she says.
Just trying to give you the skinny on adrenals and thyroid hormone treatment and the contraindications you are referring to.