For what it's worth, I am a licensed vet tech, (worked for 10 years and now I'm home with my child). I also recently put down my 18 year old cat due to quality of life issues, and got a big guilt trip from a veterinarian about it.
My personal standard for testing is this: Will the test change how you treat the condition? If the results of the test will impact what you do, then you should do the test. And if you run the test, can you afford the treatment that the test now indicates? There's no point in running a test to diagnose a condition that you can't afford to treat.
My question for your vet would be "What diseases are you screening for? What is the treatment plan for those diseases? What is the prognosis for a pet of this age with that disease? What kind of quality of life will they have? What is the prognosis if we *dont'* treat the disease?" The vet would have to present a very, very strong case that surgery would benefit my pet for me to consider it at age 14. Something like "The pet has a large mass that is causing limited range of movement. If the mass is removed, they will be more comfortable and will be able to walk". Something like that I would probably do, (or else consider euthanasia). There are probably other reasons for surgery at that age, but I'd have to know more information about WHAT the doctor was looking for and WHY they want to do surgery. Like...what's the end goal here?
Honestly, for a 14 year old pet, I would tell the vet that you are interested in comfort care/palliative care only. They may still want to run some bloodwork to make sure the kidneys and liver can tolerate the arthritis meds, and I think that is fair.