My recollection is that his cancer management was botched. He reported throat soreness, the German doctors thought it might be cancerous (though treatment would render him mute if found), he visited an English specialist who denied it was cancer at all. He got worse. After a year or more of gargling, resting the throat and taking baths, new specimens were taken and the English doctor agreed it was cancer. The old Kaiser (Wilhelm I) vehemently urged that he not have surgery (because it risked death and guaranteed mutism) and Friedrich (possibly in denial by now) decided to put up with the pain and either recover or die eventually but able to perform his duties in the meantime. In the end by the time he became Kaiser he was clearly dying and unable to speak.
Had he been properly diagnosed in the early stages, as someone already noted, my understanding of the medical technology of the mid-1880s is that he could very well have lived a decade longer or more (taking into account if they cut out all the tumour/s before they spread he would still be susceptible to new tumours elsewhere ie remission and relapse / recurrence) but with a tracheostony and therefore unable to perform vocal elements of his duties. I get the feeling the ethos of the time was if he can’t perform all the functions of an Emperor and shows weakness then better he not rule at all. Though this may just be his conservative enemies considering him too influenced by his ‘English’ wife and happier to bypass him for his son. From what I can see this ethos is also reflected in his son Wilhelm II with the hiding of his palsied arm and his braggadocio to present the image of strength.