Removing emissions will usually make the "bark" go away, it opens up restrictions in the flow path. You won't hear it on a stock setup unless your tuner pushes your boost to the point it builds up and has no where to go except backwards. So for a stock setup, with a stock tune, no this should not happen. The muffler and cat will not cover up turbo barks like that. It comes down to the phenomenon that I described above.
Deleting (which is a no-no these days) is not done to make the turbo bark. That's like saying you're going to remove components to make your turbo have a shorter life. Removing emissions comes down to eliminating some of the problems they brought, especially early on. Recirculation and coking up intakes (just look at threads on here and clogged intakes), regens, etc. Deleting, in theory back in the day, was making the vehicle more reliable. That's not the case with today's emission components.
It's not something worn that causes this. It's just physics. You have too much boost, when you drop the skinny pedal, it has no where to go. It happens more so on modded vehicles that can run too much boost for the components that the person decides to pair with the turbo of their choice. Sure you could tune your stock car and make your little VNT15 bark all day, but you'll have bigger problems if you're spiking boost hard enough to make your OEM turbo bark like that.