Firstly there is a "liability factor" with answering this question, so insert disclaimer here!! Always seek a professional and a 2nd assessment if desired. There are procedures and specs with worn steering parts. More follows.
Bottom line if a tie rod end (There is an inner and outer for each side) "Breaks" (Emphasis on Breaks) IMO there will immediately and abruptly be 'No' steering within milliseconds. Of course, it will be centric to conditions and the forces of physics at play within those milliseconds - seconds??. However, the vehicle 'may' steer toward a direction not by the driver, but that is the forces of physics.
Bear in mind with worn steering parts - tie rod ends, there is a loss of steering effectiveness and efficiency. The worn component(s) allow movement or the wheel itself will freely move side to side. How much is to be considered contrast acceptable tolerance.
There are specs for tolerance(s). That is measurable by a professional. Out of spec causes more rapid wear of all the steering components. Also, alignment specs most likely will not be held within spec causing accelerating tire wear, lower gas mileage, and other factors.
It can be demonstrated to the vehicle owner. However, at issue is policy regarding shop safety for any of the following. One is visually seeing side-to-side motion of the wheel. Another is actually allow the vehicle owner to perform that procedure both seeing and feeling. Visually seeing movement of the steering linkage related to the wheel under the car - safety issue!. Perhaps on alignment rack showing actual reading changes for toe-in and toe-out related to specs as a last resort = expense.