All kinds of "shotgun testing" (i.e. indiscriminate testing for everything like you recommend) have been studied, and proven worthless at best, and more often than not, actively harmful.
First, there is the issue of tests' limitation, and extremely low predictive power. For instance, if testing positive on A makes its 20 times more likely that you'll get B, and the prevalence of B is 1 out 1000 000 in the general population, your own personal risk remains low enough that nothing has changed -- except that you will panic and do unnecessary interventions to reduce this risk. That is precisely the reason tests are asked when you already have symptoms, so if you're pretest probability of disease is 10%, a positive test results means you most probably have it, and it is worth doing something about it.
Second, all known treatments (including "preventive" ones) carry non-zero risk. When you don't have symptoms, whether or not you test positive, your risk of dying from a disease remains lower than dying from an intervention to prevent the disease -- thus, you gain nothing by testing.
Let's take for instance a 40-yo female who gets an ECG done for no good reason. It shows signs of heart disease, which could be a variant of normal, or a sign of a disease. The lady is worried, so she goes on with a stress test just to be sure. She tests positive (a sizeable proportion of those tests are false positive for multiple reasons), so she decides to go on and follows up with a coronary angiogram to see if there's any blockage. Angiogram is normal, but, a coronary is perforated during procedure (1/10 000 risk), and she dies on the table, when she never had any health problems beforehand. This kind of stuff happens all the times.
Finally, from an ethical stand-point, as long as healthcare -- and the individual's stupid testing choices -- are paid for collectively, individual choices should be severely restricted.
If we were in a country without any kind of state-sponsored healthcare, where you'd get to pay for any self-harm from your own pocket, I'd argue for free-for-all testing for anyone without any oversight.