I'm sorry to hear that your pup Verdi has functional thyroid cancer (meaning that the tumor is producing hormones). Thyroid cancer in dogs is very rare in and of itself, and most tumors are NOT functional tumors.
Please be rest assured that veterinary teaching hospitals will never recommend substandard treatment. Their primary goal is to teach veterinarians that will be out in general practice. Thyroid tumors are not something a general practitioner would usually attempt to remove surgically. Given their location, with many essential nerves and blood vessels surrounding them, these types of surgeries would be referred to a board certified specialty surgeon. It would serve the students best to learn proper protocol based upon what is best for your dog. It is essential that these students learn to formulate a plan to give their patient the best outcome. It would not be prudent to teach them the wrong protocol.
Does your dog have evidence of metastases now?
Perhaps if she doesn't have metastases now the thought is to remove the tumor before the tumor has a chance to metastasize, and to try to minimize any further weight loss.
Treatment for thyroid tumors can be surgical removal (or debulking) of the primary tumor alone or more commonly combined with radiation therapy and possibly chemotherapy.
Once they remove your dog's tumor a pathologist will check the edges of the removed tissue to look for clear margins or evidence of tumor cells at the edges. Further treatment will be based upon whether the edges are clear of tumor cells and what type of tumor she has.
Best of luck with your girl. Please let me know if you have any further questions.