I didn't understand the question.

1. How will a high school student be able to pursue degree studies at the Open University in difficult subjects if he hasn't had experience with exams at the matriculation level?
2. What about those who won't continue to university? Isn't the system responsible for providing them with a minimal education?

Let's divide the subjects into essential ones and tactical ones. Let's say that Hebrew Bible is studied in high school in order to know the Hebrew Bible, while mathematics is for sharpening thinking. So—

3. Isn't there importance to studying the essential subjects in order to gain a minimal body of knowledge, assuming students won't encounter those subjects through the Open University—for example, Hebrew Bible and literature?
4. And regarding the tactical subjects—is there another way to sharpen thinking and analysis? Maybe teach chess instead of mathematics? But since we'll need operational measuring tools, then once again we've arrived at some kind of exam.

I didn't understand what the claim is, and what the alternative is.

This discussion is bizarre, and I don't understand who is arguing with whom. I said that education and tools for thinking should be provided. I added that regarding the question of what the method should be, and whether this must be done through matriculation exams—my answer is: not necessarily. It's really not that complicated, and I don't see the slightest connection to everything you added here.
If you expected me to lay out a full pedagogical doctrine here about how one imparts education and thinking skills—that isn't what you asked (you asked whether matriculation exams are mandatory), and this isn't the place to do that.
And if you're talking about university admission requirements, that's an entirely technical question. Call and find out what the admission requirements are, and if you need matriculation exams for that—then take matriculation exams. What exactly is the discussion here about?