The problem with this argument is that all four of its premises are assumptions: They are based on a specific idea of what god is presumed to be. None of them are necessary for a god to exist.
That means that rather than saying god is omnipotent, god doesn't like evil, god could and would eliminate evil, evil is still here, therefore, god does not exist, we can only say if god is omnipotent, and god doesn't like evil, and god could and would eliminate evil, and evil is still here, then, god does not exist.
If any one of the four premises is invalid, then the whole argument is a logical fallacy.
Is god omnipotent?
That depends on your definition of god. What do you mean by the word god?
Perhaps an all-encompassing definition that would cover the various images and ideas of different belief systems is something (or some things) or someone (or a group of persons) that is (are) outside our own universe and has/have somehow caused this universe.
Sure, Judeo-Christian-Islam religions tend to think of a single person (or a trinity) that simply wished this universe to be and it was. This person (or trinity) is generally considered omnipotent. So, the first premise will hold true for the JCI tradition.
But there are other traditions as well as modern views. It is, for example, perfectly possible, and believed by many, that our universe is a game in virtual reality. A game we have played for so long we have completely forgotten it is a game. A variation is the belief that this was an experiment in virtual reality, an experiment that got out of hand.
In this view, this reality was built by a person, or rather a group of persons, who is/are definitely not omnipotent.
Others view god as a force, as something impersonal. In that case, not just omnipotence but the entire argument is moot.
Doesn't god like evil?
This, again, is mostly a JCI issue. They tend to believe that god is good and hates evil. But the concept of god does not require the god to dislike evil.
It is perfectly conceivable that there is a god and that this god not only does not dislike evil, but this god is evil. This god may have created us to be its slaves, or just a source of its amusement.
In his Far Journeys, Robert Monroe claims we have been created by some advanced race living in a different realm of existence (thus, fitting into the above definition of god). They created us as the source of energy for themselves. Worst of all, the more we suffer, the better the energy we produce. Dang!
Could and would god eliminate evil?
Again, that depends. The JCI god could. But not necessarily would (the JCI believe in free will).
The virtual reality god either might but cannot (the experiment gone haywire), or can but would not (the game theory, in which evil makes the game more challenging/interesting).
And the Monroe race? Whether they could or not, why would they want to? The more we suffer, the better for them!
Is evil still here?
Many years ago, I heard a story. It was a very long story, but I only remember its beginning:
A young man lost his arm. The loss was an evil thing: Unlike his friends, he could not get a decent job and make a nice living.
Then his country was invaded by another country. All young men were drafted and had to go fight the war. Luckily for this young man, he was missing an arm and not suitable for military service. All of his friends were then killed in the war. The loss of his arm was a good thing—it saved his life!
I do not remember the rest of the story, but it alternates examples of why the loss of his arm was evil at one point, good at another.
This makes the concept of good and evil quite relative: It is how we perceive matters that makes them good or evil.
Even should we accept the JCI concept of a personal god who is absolute, omnipotent, etc, then we would have to admit that we could never understand what goes on in the mind of such a being. Just because we, in our relative world, consider something to be evil, we could not say that this absolute god thinks of it as evil.
I am not saying that god exists. Nor am I saying that god does not exist. I am only trying to show that the existence of what we call evil does not constitute a proof of god's non-existence.