The problem of universals arises because we generally encounter things in the world under kinds and properties. As in, I go throughout the world and see apples, cans of coke, trees, people, yellow bananas, tall trees, square boxes, grey elephants, etc.
The problem of universals can be addressed in two ways, either we need to explain what these universal categories are or we need to explain away the way that our cognition works.
Realist Explanations
The two most famous classical Western accounts provided related but different explanations, with Plato positing the forms/ideas (from a separate world) and Aristotle essences/forms (in natural things as their organizing principles). This manner of thinking was echoed in Augustine and Aquinas.
Modifying your (1):
Plato: "This apple and this ruby are both red because they participate in the form of red." (1')
(though my memory seems to indicate there's a problem with colors as forms for Plato but we'll ignore that).
Aristotle: "I intuit that that this apple and ruby both have redness." (1'')
(though for Aristotle redness is not an essence)
Traditional Nominalist Explanations
But later medieval thinkers like John Duns Scotus and John Buridan shifted to views we call "nominalism." They thought that the answers to how these sort of predications work are in the names of things and in our minds rather than in things or off in a Platonic heaven. For the most part, though, it's not that they deny a category of man that exists in the mind, but rather that they deny that the category has a necessary existence founded outside the mind.
These views do not postulate a thing for the universal -- they take it to be a concept in our minds (addressing part of your concern in a comment)
Medieval nominalists: "This apple and this ruby both fit with the word/concept red that I use to predicate" (1''')
OR
""This apple and this ruby are both red, because this apple fits under the predication of the name/concept red and this ruby fits under the predication of name/concept red." (2')
(where predication is understood as the mental activity of applying names to things).
(There are of course people who denied there were existing essences both in the West in some forms of skepticism [elsewhere too] and in the East).
Descartes too has a philosophy that depends on the nature of ideas in our head but takes these ideas to have an origin outside the mind (think for instance of his argument for God's existence in Meditation 3 which hinges on the origin of the idea of infinity and thus has both ontological and cosmological elements).
The Problem with Universals
There's all sorts of problems with universals, a problem noted as far back as Plato's Parmendies and what Aristotle called the third man argument where forms multiply endless. There's also a problem of where forms come from -- in Plato's case, they are all stored up in an unchanging "heaven." In Aristotle's case, the essences wind up being interlinked with an argument for the immortality of the soul.
The Problem with Explaining Away Universals
In response to the problems with universals, several philosophers in the logical positivism / philosophy of language stream have as you noted taken up views like "predicate nominalism." These views attempt to explain away universals. But this is done at a cost, viz., as you note you have to make predicating "unanalyzable and primitive."
While this gets you out of explaining how universals work, you've now asserted we cannot explain what we do when we predicate. I called this "naive nominalism" for two reasons. First, insofar as we've lost the mental apparatus that the earlier nominalists used to explain how we predicate. Second, this is a mirror of what Hegel calls "sense certainty," the view that truth is just what I see when I see it.
In this case, the problem of predication would be justifying why we've given up on explaining predication but still think it's somehow meaningful. To make this move work, you're going to need a pretty substantial defense of the claim that predication is unanalyzable that still allows us to (a) make negative predications and (b) allows us to predicate of multiple things.
You ask:
Why can't we use the predicate 'red' without it and take it as unanalyzable, primitive? "The apple is red." just describes how the apple is.
You seem to be saying we cannot know what we mean when we call the apple red. That strikes me as quite strange. I think we mean that an apple has a certain color. Or more generally, it sounds like we are asserting some object has some property. But you're telling us we are doing something we cannot understand.
A second related issue might make it clearer. It seems we can also make negative predications: the apple is not green. But is this too to be seen as unanalyzable? If so, how? It seems we are saying, this apple does not fit with this property.
When you state (2),
"This apple and this ruby are both red, because this apple is red and this ruby is red." (2)
The problem here is that if [this apple is red] and [this ruby is red] are unanalyzable predications, then it's not clear how we can break the [] and take the word red and apply it to both. Maybe, you're saying we cannot.
If so, then you're saying we cannot compare things, which seems a long way to go to avoid every possible formulation of universals. If you're saying we can, then what is the red that's detachable from particular predications in this way?
A third possibility is you're saying that the "is" of predication is difficult. And it is. But then that seems to revive the problem of universals, and to just be saying that's hard to know what these predicates we seem to work with are.
To sum up, if you're trying to avoid "universals" to avoid an overly complex metaphysical world, that's an admirable goal. But not all views that include "universals" are committed to saying they are independently existing entities, and if you have to say we cannot analyze predication avoid them, then that invites some pretty deep problems of its own.
Another issue that you raise is the types of predicates. I haven't addressed that very directly (only by mentioning that Plato and Aristotle deeply limit the number of entities they are dealing with or at least notice the problem), but clearly there are nonsense predicates (made-in-1816-ness) that no one grants are universals. We needn't say anything about whether there are any things like universals inside or outside of minds to avoid these.
