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Y'all ever stop and think about how the actual first plague doctor masks weren't actually DESIGNED to look like a bird face, they just were engineered for "efficacy" in such a way that they looked like they had a bird beak.
Any plague doctor masks that WERE intentionally designed to look more beaky were likely made after hearing description of this original mask. Kind of like that drawing of a rhino by that guy who'd never seen a rhino.
This implies that the original mask design was effective enough in preventing doctors from contracting the plague that word of it's design spread. Those who heard about it would certainly latch on to the "bird beak" aspect the most. Kinda stands out.
And it likely was effective! The Bubonic Plague was the main one active at the time, which is spread by contact with infected fleas. The total coverage of the mask would act as a physical barrier against some fleas -
and depending on the fragrant herbs used in the "beak" the scent could have also inadvertently acted as an insect repellent. Granted the masks were designed in line with the Miasma Theory, that "bad smells" are what spreads disease. So it just... Accidentally worked. 
The original mask (here's the pic again) is not that bird-like, definitely compared to the white raven style ones we normally picture. The "beak" on this was constructed to funnel air through two small holes at the end a cone of fragrant herbs to remove the "bad air."
Whoever designed it likely went with this cone design for structural and function reasons, probably after testing different methods of sealing the herbs against their face for maximum air filtration. It's very likely they tested shorter, more blunt designs and breathable fabrics-
But likely found this leather cone with herbs packed in side reduced bad smells the most, probably because of it's length and that you can likely pack herbs into it tighter than other shapes, causes the air passing thru to have the most surface contact with the herbs.
Essentially creating that same friction/obstacle air filter effect that is what makes N95 masks most effective. A less direct air path means more obstacles for particles to run into and more essence of the herbs that is picked up.
However the people who heard about this mask and reproduced it's design took the "I'm doing the right thing for the wrong reason" thing a step further. The evolution of the design implies they believed that the bird aspect was an intentional warding feature, not a coincidence.
If they had understood either the insect repellent/barrier aspect or at least the air filtration, the design likely would have been improved on to be more efficient/simple, rather than the aesthetic improvements they made instead.
However, despite basically just making them creepier, they stuck around. That's because, as shown by the introduction of the surgical mask, some PPE is better than no PPE: a 50% reduction in spread of infection is significantly better than 0%.
So because this successfully, even if inefficiently, reduced the death rate of plague doctors, we are now blessed with this super goofy image of medieval medicine.
If you're wondering why I just took a deep dive into pondering plague masks look at the date
