How is rare steak made safe to eat?
First, 145 °F (63 °C) and higher is the temperature for a well done steak. So, with the addition of carry-over cooking, your results don't surprise me.  If you are shooting for rare, cook to an internal temperature of 125 °F (52 °C), and let your steak rest 10 minutes before slicing.
While the USDA correctly and necessarily provides temperature guidelines, in fact the reduction of pathogens follows a logarithmic curve and includes the variables of temperature and time. That means, in general, that longer times at lower temperatures will reduce pathogens. This understanding is the basis of sous vide cooking, for example.
Additionally, we generally assume that any potential pathogens are only present on the surface of whole muscle cuts.  So, again, in general, achieving the target temperature on the surface eliminates the threat.
Finally, the quality and handling of the raw product is critical.  It is important that you have fresh products, kept refrigerated or frozen until use, and handled by people who are practicing safe handling procedures (washed hands or gloves, ...).