I think platonism suffers from further issues than the video states. Take the issue from different angles:
The problem of Induction, Causality, and reliability of our minds:
It is hard to see that there is much a reason to suppose that an atheist affirming the existence of abstract objects has provided a solution to this issue. Why suppose that the world of material particulars acts in a manner conducive to human faculties because platonic objects exist?
Take the reliability of human minds, why suppose that humans developed to be rational agents because universals are platonic objects? There seems to be no connection between these ideas from what I can see. There is much more work that needs to be done to show that the platonic scheme is a competitor. I think these types of issues with being systemic as long as the world of rationality is impersonal separate and almost unrelated to the matters of this world.
The person may argue that the world gains its character from the rational forms and thus act completely rationally via they are instantiations of universals. The issue becomes whether the world actually is an instantiation of such things. How does that person know such a thing? Why think the world is in accordance with those properties rather than the properties of chance, chaos, and madness?
Universals:
The issue with platonism is an issue of content. What does it mean to be an object in general with no particular features? If you were to think of tree-ness, then how would you describe it? You might think back to a tree that you’ve seen before. It probably was brown, with several branches, various colored leaves at certain times of the year. The issue is that trees vary in color, they often have different numbers of branches, and they don’t always have leaves. So, the universal purely abstract tree tells us nothing about the particular trees we see. You experience these attempts to be purely abstract all the time. For example:
It is like when you and your significant other wish to eat and so you ask them what they want and they say “Food!”. It becomes too generalized that it tells you nothing about what you want to eat.
Ethics:
In recent times, platonism has been a place where people have gone to try to show that moral realism is possible on atheism. This has been the move by philosophers like Erik Wielenberg. There are several issues with this move. This is also interrelated with some of the prior objections. Take the issue of them being universals:
In Euthyphro, Socrates argues that piety cannot be defined as what the gods desire. For why should they desire it? They must desire it because it is good. So piety is a form of goodness, and goodness must exist independently of what gods or men may think or say about it. So it must be a Form. We should note, however, that if courage, virtue, goodness, etc. are abstract forms, then they have no specific content. To know what is good, for Plato, is to know the Form of Goodness. But Goodness is what all individual examples of goodness have in common. How, then, does it help us to know specifically what is good and what is bad? Any time we try to define Goodness in terms of specific qualities (justice, prudence, temperance, etc.) we have descended to something less than the Form of Goodness. The Form of Goodness serves as a norm for human goodness, because it is utterly general and abstract. Any principle that is more specific is less normative, less authoritative. Such is the consequence of trying to understand goodness as an abstract Form rather than, as in biblical theism, the will of a personal absolute.120
John M. Frame. The Doctrine of the Christian Life (Kindle Locations 2232-2241). P & R Pub.. Kindle Edition.
The other issue to discuss is whether a platonic form can obligate one to act. It is far from self-evident that we should follow them. If we are not obligated to concrete impersonal objects then why think we are obligated to abstract impersonal objects?
The other issue is the ultimate motivation for our actions. On some platonist’s schemes, there is no afterlife. There is also no judgment for our actions and no reward for our deeds. It seems this leaves a person no reason to do a good thing given that it makes no significant difference to the development of the world.
Epistemology:
There is a difficulty as to whether we can actually know these truths. If you are familiar with Plato then you may recall that the forms are unlike the things of sense experience. They were so unlike the world that we only know them through recollection. The forms were known from our past life with them.
This is where some have posited self-evident ideas or things known by human intuition. The issue with this appeal is that it leaves a person in subjectivism. People have intuitions about different ideas all the time. Why should we prefer one’s intuition over another?
the other issue is that we suffer from human finitude. We don’t have a comprehensive knowledge of everything that exists. That means we may always be unaware when trying to posit a universal on a platonist scheme because you will never be able to know whether there are further universals needed to understand some given object. This is clear from thinking about universals as a hierarchy. For example, we may think about “dog-ness”. There are universals or classifications that “dog-ness” participate in “mammal-ness”. That also goes further is participating in many other universals. Since we may be unaware of an infinite many of them, then we never can state that another universal doesn’t exist that contradicts how we understood any other universal such as “dog-ness”.
If man is simply matter in motion, then his thinking apparatus (the brain) must be characterized by what is true of matter and motion. If his brain is finite, particular, and changing, its operations or results will not manifest the opposite characteristics (pertaining to truth, laws, principles, etc.) of universality and immutability.
Greg L. Bahnsen. Van Til’s Apologetic (Kindle Locations 9699-9700). Kindle Edition.
Further Suggestions:
The Council 
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