"I use both approaches just about every day" is the only sane answer here. Different tools for different jobs. What would think if you met carpenters who described themselves as "Hammerists" or "Sawsallitarians"?
Sure, but there's nothing which would prevent you from addressing that from a Bayesian perspective. In fact, Bayesian particle filtering techniques would probably be a great tool for "on-line" quality assurance.
I'd think they just made it clear how to choose one or the other based on whether I wanted something built and/or destroyed, as opposed to whether I just wanted it cut apart. ;)
I'd say I'm philosophically Bayesian, but frequentist techniques are often more convenient.