Why aren't bird radars placed on the front of commercial aircraft?
There really is no such thing as &quot;bird radar&quot;.  Radar comes in a few frequency bands that have certain advantages depending on the distance, resolution and attenuation you need out of it.  The band isn't the only factor here as the band will dictate what size dish you need and your aircraft nosecone will but a distinct upper limit on that.  Weather radar is geared toward detecting rain drops and anything bigger than rain (hail, bugs, birds, etc) with reasonable distance and attenuation features (you want to penetrate into a storm).  Birds will show up on this radar, though depending on smoothing and processing algorithms you may need a flock of them before they get drawn on the radar, at which point they'll look like some rain.  A radar optimized to detect bird sized objects would probably use a wavelength that did not let you see things as small as rain drops and small hail.
Some other things to keep in mind are that the radar is not always on, and it should be off on the ground except during takeoff when needed.  The reason for this is that you do not want to be putting radar energy into people, and they tend to be in the way on the ground.  We only turned the radar on for takeoff when there was significant weather around the field and wanted to be prepared for requesting diversions.  Aircraft radar tilt can also be adjusted, so we wouldn't be looking at birds in front of us, but clouds above and in front of us.
Wit that said, our SOP on fields with known heavy bird activity was to takeoff with the radar on and pointed forward because someone had the idea that the birds didn't like being in the radar beam.  I have no sources to back that idea up, but it was written into our FOM.
Now assume you can takeoff and see a flock of birds on the radar.  Do you have enough time to do anything about it?  If so, what do you do, some maneuver away from them or do you turn into them in hops they take the evasive action?
Now back to the title question: Why aren't bird radars in the nose cone?

The weather radar can see birds, it would be redundant.
A bird radar (optimized for birds) would not be good at seeing weather, so where would the weather radar be placed?
In a real situation, what would a bird radar enable you to do with a positive detection with the time you have to react to it?  Probably not much.
Weather is a much more common threat than flocks of birds, so if you can only have 1 radar system, you want the weather radar.