The history of blood glucose testing is informative here. It used to be a hospital lab test, like most other tests: done fasting, infrequently, to diagnose diabetes. Today, that test is done with an over-the-counter test kit, and diabetics do it multiple times per day. This provides information that the infrequent, hospital version couldn't provide at all: how blood glucose responds to meals.
There are many other tests which, if they could be frequent and self-administered, would enable people to make new discoveries. The common metabolic tests, for example - HDL, LDL, etc - are very closely analogous to glucose tests, in that they respond to meals and that response is probably more informative than the fasting test. But there's nothing analogous to a glucose tolerance test for cholesterol, because that requires ten tests in a row and that's too expensive.
One of the more common complaints I hear from friends is about migraines. Blood tests are worthless there because it's impossible to get a blood draw during an actual migraine. Same for most mental illnesses; comparing blood tests between a bipolar person's manic and depressive phases would be fascinating, but no one does it.
And that's not even mentioning micronutrient status screening. The rates of micronutrient deficiencies in the United States found by the National Health and Nutrition Examination Survey (NHANES) are shocking; it's the twenty-first century and the rate of iodine deficiency is 9%.