In theory, that's a really good idea. As fulfillment I really like the possibilty. For example, let's say there is a Circle deal going on Pillowfort so people are ordering the cheap ass kid's dinnerware in kitchen. (I'm hitting real close to home for fulfillment peeps.) Well, if the kitchen DBO sucks, falls behind in frieght, never scans his outs and lows that makes for a really bad week for fulfillment. If someone orders 10 blue plates that we're supposed to have 37 of, but that we haven't sold in 50+ days and we don't have any, then why should that be MY problem. Yet, it counts against my metrics. If the kitchen DBO had to pick that order, it becomes their problem. Now, they have incentive to do their job right and complete all their routines on a regular basis. This should lower INFs.
But, that's all in theory. In reality, even the best DBO can't get everything done and done right all the time. Even if they could, they can't be in the store all the time and always know what happens when they aren't there. That's where fulfillment experts come in. We know how to find stuff when things aren't done right. That's what makes us "experts." It's why we're needed.