As a logician, this hurt me on a personal level. As a philosopher, I suppose it is good to see skeptics. As this closely ties to my field, I will not hesitate to chime in.
First, let's look at the semantics. You essentially answered your own question:

So is math real? Is it a fundamental part of the universe or a man-made concept?

You inquired whether math was "man-made" or "discovered". In accord with your own video, you can only discover something that is there, meaning it existed. OR, it is man-made, meaning it was developed and crafted. You cannot develop and craft nothing, and therefore you would have something. Something being maths, and very much real.

Now I think you meant the question of whether maths is real as in physical and material, or whether it is just an abstract concept. In each school of thought you examined, it is easy to see who attributes mathematics to physical properties and abstract properties. As you've also discussed, physicalism is full of holes and mathematics is not empirical like the other sciences.

I will go ahead and proclaim that mathematics is "abstract".

Many of these issues were discussed by Frege, but no mention of him in any of these videos, which is almost heartbreaking. Frege gave one of the strongest, formal definitions for number in his Begriffsschrift and The Foundations of Arithmetic. As we know, mathematics is a foundational field. It builds upon itself ie. multiplication is a series of addition, etc. However, prior to Frege, no one really offered a good definition for the basic constituents of math, number. As you said, there were theories like physicalism and psychologicalism, but these were all flawed in one way or another (Frege explicitly points out their flaws in his texts).

Bertrand Russell, who was also ignored in all of these videos, made slight, but important revisions to Frege's definition in Introduction to Mathematical Philosophy (primarily Chapter 2).

Here is a brief summary of what they say, but anyone that is interested in the philosophy of mathematics should read the texts directly:

1. A class is a collection of members
2. Suppose some class α is defined as "finger on @samueldouglas's left hand" and a different class β is defined as "toe on @answerswithjoe's right foot".
3. Two classes are said to be similar (equinumerous) if and only if each member of class α has one and only one correlating member in β, and all members of α and β have a correlating member. With the example in [2], α and β are similar.
4. If we then encapsulate all similar classes in a class, ie. all classes with only two members (pairs) in a class, we arrive at a class of classes.
5. We can define the class of classes to be the number itself, ie. 2 is the class of classes with only two members.

If we define number this way, we can see how it is abstract, namely that it is a class of classes. As classes are in no way physical, they are abstract. Once again, note that this is a brief summary of Frege and Russell's definition for number, and not their entire argument. We should also note this is not a perfect definition, see Rusell Paradox. This can be solved in terms of Zermelo-Fraenkel Set Theory and other methods. However, it does provide a good basis for the definition of number, which is foundational for mathematics.

And returning once again to semantics, even if mathematics were abstract, it would not make it any less "real". Take the abstract concepts of happiness, freedom, or motivation. Those are no less real than an abstract concept like maths. Questioning the existence of those is an entire other field of philosophy which is not adequate for this discussion.

Lastly, I in no way intended for this to be an attack on your discussion. It is an interesting one and has been thought about for thousands of years. However, I saw a lack of Frege and Russell, who both made groundbreaking work in this philosophical inquiry. I believe mentioning their names is necessary in discussions on this topic. Thanks for the post.