My short answer would be no, careers of people who study pathogens requiring biosafety training are not focused biosafety level.
More broadly, the people I know who study infections diseases are not motivated by the biosafety levels of their infections organisms. Rather, they are motivated by biological questions and their research that often require higher biosafty for their labs.
For example, the researchers featured in a recent (July 2015) National Geographic do not study Ebola because they want to work on a BSL-4 level virus. Rather they want to about learn about the disease from both a basic science and applied human heath perspective.
Your hypothetical Dr. Kliener would not work his way up the biosafety chain as a conscious. Let's say a young Mr. Kliener always wanted to study human health. He probably would have started doing research in a BSL-2 facility as an undergrad research assistant (See BSL-2 requirements here. As a grad student, he might have worked at a BSL-3 lab if he choose a good program while spending years focusing on a specific pathogen or system. By the time he became because a PI, he would have needed to find a BSL-3 facility for his research. He worked his way up, but not because he wanted to study dangerous pathogens. Rather his research question guided his path.
Other career tracks could end up using a BSL-3 facility, but not necessarily need the stop at a BSL-2 facility. For example, a hypothetical Dr. Fish may have started off loving the outdoors. She studied limnology as an undergrad and grad student and did not step foot inside a lab. However, when she began working as biologists for a state conservation agency, a novel fish virus emerged. She now is tasked with studying it. Her agency trained her to work in a BSL-3 lab so she could culture fish with their virologists.