In that situation, only two kinds of test will avoid false positives: (1) a perfectly reliable test, or else (2) a broken tests that never reports a positive.
Suppose we have a perfectly reliable test for tuberculosis, but we foul it into having an equal false positive or negative rate as follows: we perform the perfect test, the flip a fair coin, and if it is "heads" we invert the test's outcome.
Then if we apply the test to a population in which the incidence of tuberculosis is low (which is actually the case), most of the false results will be false positive.
In other words, a high false positive rate in a screening test relative to a low false negative rate does not prove that the test is biased toward false positive identification.
Consider that if nobody at all has tuberculosis, the only kind of false result possible is a false positive.
I think that they are saying three things 1) real tests aren't actually trying to minimize both types error of which is the opposite of the hypothesis "It might be natural to conclude that physicians seek to minimize both types of error" 2) illustrating with a very dangerous disease (to give the reader an idea of risk) the difference in types of error. 3) Demonstrating potential trade offs that occur, Type 1 error is so much less risky they are ok with 50x as much error.