Streptococcus species belong to the normal flora of the mouth. They can be cultured easily. Most common are S. sanguis or S. mutans (which causes caries). They probably get in the bloodstream on a regular basis by little injuries during tooth brushing. s. pyogenes that causes strep throat does not colonize the oral cavity.
I never heard of a doctor checking your teeth to see if you have strep throat. Normally, you would check your throat and the signs are pretty obvious. However, maybe new technology can catch the strep throat easier.