Depends on what you want you want to do with Bayesian data Analysis. If you want to be a mouse mover and button pusher, then yes perhaps you can get away without knowing the math, but even then I doubt that's true.
A job of a data analysis also requires one to adapt to the new situations, variations etc. The job is closer to that of a car mechanic than that of a car driver. Its not really a brick laying kind of a job.
Knowing the math is not a all or nothing deal, one need not get into its most sophisticated avatar right from scratch. However, if one one thinks that one can become something more than a mouse clicker, the person is just deluding oneself.
Why would somebody want to become an airline pilot without doing a graduate degree in fluid dynamics?
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Not everybody needs to drill down to the most fundamental levels of a discipline.