Blood can transmit. Blood contact very rarely can transmit syphilis. It probably requires much more blood than possible during sexual exposure, shared needles, etc, but transfusion can do it, especially with non-refrigerated blood products. It's extremely rare and even without testing, probably there would be few if any cases, at least in the US. OTOH, testing is easy, cheap, and ingrained by history and habit.
Real reason. Cooling of blood in the blood bank kills the spirochetes, but fractions such as platelets that are kept at room temperature could theoretically transmit the infection. The test is easy and with all the worried people around, wouldn't you want donors checked even if the risk is miniscule?
Easy detection. Testing for syphilis is done through blood as this is an easy way to detect it. You are correct that it is passed from person to person via skn to skin contact with an infection sore called a chancre. Once the host has syphilis it can be easily detected through a simple blood test.