We're in the realm of probability, which is mysterious sometimes even to people who have a math background. I didn't fully grok it until I studied intensely for my first actuarial exams. (:
This kind of thing is almost always a weighted coin toss: with sensitivity or specificity alone, you only have two possible outcomes (present/relevant, absent/irrelevant), and the thing that changes is the probability distribution of those outcomes.
Combining the two gets you the full four: present and relevant; present and irrelevant; absent and relevant; and absent and irrelevant. Taking the uniform distribution, they're all 25% likely, but the idea is to find a probability distribution that makes the "present and relevant" and "absent and relevant" outcomes more likely.
Since I myself don't work in a clinical setting, I simply hadn't considered that the clinician would want to exercise discretion in pre-screening for specificity before ordering the test in order to get there. Oops.