"No birds, no animals, nothing"

It's not explained in the film, but it is eventually explained that less than 10 years ago -- since 10 years ago was the disappearance of the Prometheus mission -- David and Elizabeth Shaw arrive at this planet and David releases the capsules of black ooze that cause the transformation or consumption/destruction of every non-plant form of life.

There is one clip relevant to help corroborate David's story to Walter, that is otherwise only partly shown during the film:

Unfortunately there's no mention of whether these events take place over months or years, but David's "work" on the planet is extensive, at least taking several months/years of eugenics, even with the hyper-fast growing species.

[We see the small alien grow to almost full size within just a few minutes after it first bursts out of the first soldier's back.]

So, biologically speaking, if similar to Earth, it should require a complete eco-system for the very similar plant-life to be sustained. However, with David's experiments, most of the wildlife is transformed into alien forms, and they may or may not be participating in the ecosystem.

Since trees live for decades, if not centuries, they wouldn't necessarily need constant cross-pollination to survive, so long as the chemical nutrients were still available in the soil, rain, and atmostphere.

https://en.wikipedia.org/wiki/Pinophyta#Female_reproductive_cycles

Since natural reproductive cycles of coniferous trees -- which many in the film appeared to be -- can take 3-4 years, only missing one reproductive cycle would not obliterate the species. The lack of reproduction itself also does not determine the viability of a member of a species, it merely affects the persistence of the species as a whole, since no new members are brought to life.

So, since everything occurring on the planet has taken place over the course of roughly 10 years, and the plant-life is robust to the natural storms of the planet, it seems reasonable that there would be no visible effect, yet.

Likewise, there is also a lack of fruits and flowers shown in all scenes in the film, and the wheat pods were brittle and decayed -- https://en.wikipedia.org/wiki/Wheat -- suggesting that the fruiting plant-life was dying. Even though wheat is a cultivated plant, if it is already growing in a fertile, extant ecosystem (per your question) it probably should have survived. It just may have overgrown or undergrown its potential, without any farming/cultivation intervention.

So while your understanding in your question is quite reasonable, in that many of the plants probably would not exist; they already existed. There has been a cataclysmic event within the past decade, but the plants were already extant in the ecosystem.

If you as a person already exist, but you only exist in the first place because of your parents and their parents and their parents and so on. And they all only got together to make babies because of red wine and the full moon. And then one day all red wine vanishes and the full moon becomes really annoying and non-romantic. All of humanity doesn't stop existing. Even within 10 years, with our 9 month gestation cycle, most of humanity would still exist, people just wouldn't be having children. So as a collective species, we would be barrelling towards extinction, unless people could start figuring out how to make babies without red wine and a full moon.

Maybe switch to Rosé and the Aurora Borealis/Australis?

"Life finds a way..."