I can’t really tell if that is a very high number or not without knowing the population of birds.

If 377 Million people were killed annually by something, would it make a difference what the population was?

It's a horrific number whether you've got a population of a billion or 10 billion.

Of course it would. About 43000 people die in car accidents in the US every year. That is apparently considered acceptable in a population of 332 million people. If the population was one million people, what would most likely be considered horrendous.

I admit that my standards regarding death of critters are somewhat different than death of people, but the total population do matter there too. 6 million deers are shot in the US every year in a population of 25 million. People don't seem to be very upset about that, but rather concerned that the population still is rising. Many people are very upset that any rhinos at all are killed.

And as the example with deers show, you can cull some populations quite heavily without any apparent ill effects.