I did field studies of birds in northern Brazil as a grad student and collected (killed) several birds that I caught. It is a controversial subject, with the viewpoints tending to be spread over a couple of different dimensions.

In South America, teaching of biology at universities seems to me to have a heavier focus on evolutionary biology, taxonomy, biogeography and systematics, and from what I've seen in Colombia, Brazil and Peru, collecting museum specimens is uncontroversial among biologists.

At the other end of the spectrum, in the UK, those areas of evolutionary biology are undervalued, with a much stronger focus on, for example, conservation biology and "hypothesis-driven" science. That, coupled with the greater squeamishness of the pet-loving garden-bird-feeding vegetarian-heavy gun-eschewing Brits (I'm one), means that collecting museum specimens tends to meet with disapproval from British biologists, in my limited experience (did grad school in the US). But it must be said, the UK does very little primary field research of that type.

And in the USA, collecting specimens is an issue which mostly does not affect universities, since most of that sort of field work is done by the natural history museums (AMNH, FMNH, SMNH, etc) with their research wings, staff field biologists, and PhD students doing field work. From what I understand, at the Field Museum in Chicago, there was a more-or-less furious disagreement between the conservation biologists (against) and evolutionary biologists (pro).

In any case, I killed a few birds with a shotgun, but mostly via thoracic compression (take a beautiful small bird you've just caught in its natural habitat and squeeze its chest hard; it will writhe and then have a heart attack and die in a few seconds). These were all small birds. It's not a nice thing to remember. Natural organisms in their natural habitat are one of the most important things in the world to me. But the scientific arguments for preserving examples of an entire body in a research collection are extremely strong and it doesn't endanger the population, obviously you don't kill a bird if there are only tens of them in the wild. So it's up to you and the battle between your personal beliefs about the sanctity of life and the possibility of some sort of consciousness in small birds on the one hand, and the objective arguments that museum specimens help science and help understand the evolutionary lineages that we are trying to conserve, on the other.

Edit:

(1) You would typically collect one specimen per species (or perhaps one male and one female) per locality, only if there were not already existing specimens of that species from that locality. "already existing specimens" = in any museum in the world; you don't collect specimens to add to one museum's research collection. Biologists get on planes and visit museums to examine the specimens there.

(2) This all has nothing to do with the taxidermy of specimens for display to the public. The birds aren't stuffed and displayed artistically perched on branches or whatever. This is collection for the research collection part of the museum: draws and draws of birds stuffed in a standardized non-artistic way allowing the body and plumage to be examined.