I'm going to suggest a "middle way". 
We've had older dogs put down when it was clear they were in pain and not likely to get better, and we didn't want to put them through difficult or invasive surgery, for instance. BUT as a broad rule of thumb, I think if you are going to have an old dog, you should be willing to consider actively managing their health, their quality of life and ultimately the end of their life, rather than just avoiding the issue.
Like other posters, I don't regard old age as a disease (my thinking was transformed on this issue by a vet some ten years ago who was passionate about reframing how we approach the care of older dogs).
As a recent example, we have a greyhound who's just short of her 14th birthday. She was already on Metacam and Nutraquin for joint stiffness but we noticed earlier this year some difficulties for her - stiffness, less running about, some dragging of feet etc.
We discussed it with our vet and made it clear that we wouldn't want to put her through surgery but we wanted to know more about what was going on. After X rays (which I don't regard as either invasive or traumatic, in general) we discovered that her hips were fine, but she had some bone spurs along her back.
She was then referred to our local clinic where she had gait analysis and the wonderful vet there prescribed a personalised programme of massage, acupuncture and hydrotherapy, plus some stretches and proprioception work for us to do at home with her. The difference has been remarkable and is a joy to behold. She has regained some fantastic muscle tone in her back legs, markedly improved her posture, runs more freely and is generally behaving like an overgrown puppy once more when we are out on our adventures.
There is nothing unpleasant or invasive about the treatment in the clinic and while I'm extremely conscious that we have a limited time left with her, I am happy that we have done everything we possibly can to ensure she has a comfortable pain free life for as long as we have her for.
I'd also add that it's essential to have a good dialogue with your vet - we are very blessed with ours who I trust and respect greatly, and we've had many discussions about each of our dogs and our approach to their care (our red lines etc). Having that foundation of a relationship where you understand each other's approach to care is vital, and also saves a lot of time if you are ever in a traumatic situation as I know our vet knows pretty much exactly what I'm likely to want. Over the years, through various PTS scenarios, we've been in agreement over the timing and I've always felt comfortable with the investigation/treatment options leading up to that.
So this is a long winded way of saying it's not a black and white issue and please don't write off ALL tests/investigations - you may be unwittingly excluding some genuinely helpful options.