The New England Journal of Medicine did a study of 599 articles and 1500 ratios between 2000 and 2005 and there conclusion was "Although some preventive measures do save money, the vast majority reviewed in the health economics literature do not.". There have also been more recent studies that comes to similar conclusions.
As for your doctors advice, of course on a personal level it's generally cheaper to catch things early but that assumes you are actually going to develop something. For those that never do, or for those cases where advanced warning is of little benefit, cost-wise, those thousands of dollars of tests are effectively wasted money. The math is simple, [(cost of testing) x number in target population] - [(cost of treatment) x number of affected]
For prostate cancer, for example, it's been estimated only 1 in 1000 affected men will be saved by preventative screening. For the other 999, as well as those who never get prostate cancer, the early screening was essentially needless costs. Because of it's rate of growth, in most cases knowing early before symptoms arise, won't affect outcomes or treatments. Of course for that one man the preventative screening is a lifesaver, but on purely economic front, it's not cost-effective.