This argument would be valid if humanity would continue all practices it has done "since the dawn of time".

We have dropped some practices and we continue with some. We no longer leave the dead to rot, we bury/burn them, and so on. We developed religions, science, etc, and we are in a different era now, our lifestyle has completely changed, we don't have to hunt, don't have to build our own shelters, and we are no longer nomadic.

I am of the opinion that `killing animals` is a practice we can safely stop now, it was a necessity at that time, but right now it is completely optional.

There are various angles to look at this. One is sustainability and another one is morality.

Sustainability: Do you think we have enough animals to feed 8 Billion people on earth meat daily? I hope you know why we had to fallback to agriculture as a source of nutrition. Why most early settlements were started on river banks?

Morality: My moral standard is: Don't kill animals for my own sake of pleasure, kill only what's necessary for my survival, kill only what is there to kill me/hurt me.

So can I "kill" plants?:
Yes (Using the term 'kill' wrt plants is just wrong, but I will continue with it for the sake of argument).

How is it morally okay to kill a plant but not okay to kill an animal?:

Let's agree on the definition of an animal. By animal, we all mean the set of (humans, pets, goat, horse, pig, lion, etc), there are no plants in this set. They are in a set called `living_beings`, which will have bacteria, viruses, insects as well (who can be further clubbed into smaller sets).
Now my moral standard is "Not kill animals" (Not 'don't kill living beings'). It is on this entire set, not selectively for X or Y, which will be hypocritical. I am applying the same level of morality to everyone in this set.
Now coming to plant-based food. First of all vegetarian food is not just plants. It is fruits, vegetables (akin to fruits), seeds, leaves, and other different parts. The plants are not always "killed" unlike when producing meat-based food (except eggs). The plants are "evolutionary hardened",i.e. built for harvesting, they don't die if you pluck a fruit (moreover they drop it naturally). They don't die when you take a flower or take a bunch of leaves (as long as you are within limits). The same can't be said for any animal.

Is the use of pesticides, deforestation, and killing of insects/rodents okay for producing large amounts of vegetarian food?:

No, I am against that but I don't see any other alternative to feed the calorific needs of 8 Billion people on earth. Of course there are other farming practices but they can't be commercialized or don't have high yields. As much as we can, we should try to eat locally sourced items to avoid carbon emissions due to transportation over large distances.

So what will be my ideal world that is according to my moral standards?:
Ideally, everyone has a backyard where they can grow their own plant-based food. If you want better nutrition coverage, keep some chicken and eat the eggs. Let the chickens enjoy their lives, doing chicken things.

Will I eat an animal if I am stranded on an island with nothing else to eat?:
Yes, at my current level of ego, I would prefer to stay alive by killing and eating the said animal.