For things like STD tests it raises the barrier and cost of frequent testing. We need a more fine grained policy than all blood tests require a physician.
They don't. You can get cheap/free STD tests from non-physician providers. You can also buy STD tests at the drugstore, over the counter. But like it or not, anyone who collects biological samples from strangers' is going to require some training and licensing.
Moreover, the interpretation of an STD test is not as straightforward as you might think. False positive rates can be high for the cheaper tests, and there's natural variability in things like antibody titers. People also neglect that timing matters -- if you test too soon after infection, even the most sensitive tests won't work. Or they fixate on one particular scary illness (e.g. HIV), and neglect testing for others that are less well-known, but more common (e.g. Chlamydia). Some of these problems can be mitigated. Others can't.
Regardless, having someone trained to interpret tests is a good idea for just about everyone. Even the cheap/free/OTC tests come with that service, at a minimum. It's a bad idea to just allow people to pick and interpret their own tests, even for the "simple" ones, like STD tests.