I've decided to scrap my previous answer to make my ranting more structured. There are a number of issues with the system, including:
The system is quite different from what we have now. It would need some catastrophic/revolutionary changes to society. There will be no natural acceptance. 'Recommendations from the health specialists' isn't going to cut it. People now do not vaccinate their children, give birth at home and gleefully ignore lifestyle and dietary recommendations.
Will to participate
Simply said, it is unclear who will benefit from this system. It ignores the wishes and the will of women completely (the wish to select sex partners, to choose the farther for the child and to guarantee the future of the child), and severely restricts most of the men. Interestingly, the power and the benefits in this system are separated. The most power is in the hands of the gatekeepers - the people who decide, who will procreate and who will not. The benefit is that of the future 'alphas'. That's not how it works. The last Utopian attempt at society where benefit and power were so separated was the Soviet Union. There, 'the Party' was supposed to guide the society for the benefit of 'the workers'. Needless to say that quite soon the Party had become the ruling class that guaranteed its own benefit first.
That is the biggest and most glaring issue with the proposed system. It needs a lot of enforcement at a lot of levels of society. You are restricting the rights of about 90% of your population for the benefit of 10% in an issue as basic as reproduction. And those 10% do not seem to have any way to enforce their privileged position. Who controls the process of selection? How exactly is the sex and procreation prohibited? Is there an operation, chemical or surgical? Who does it then? What about the other countries? Is this system in one country only or a worldwide one? (Good luck having the whole world agree to something so inane) Who can prohibit a woman from going to a different country and getting pregnant there? Or giving birth there?
Even if you manage to get the system started, you still will have a lot of issues to deal with.
Gatekeeping and Corruption
I've touched on these points already, but it bears repeating. First, there is no 'scientifically best' genotype. There always is a vague societal consensus on quite self-contradicting and broad ideal. So, humans would need to implement the selection. And those humans have the most power in your world. They will be interested in playing the system for their own benefit - plain bribes or blackmail, the desire to make their own child 'privileged', the desire to hurt others, etc.
Toxic Levels of Competition
If selection is done later in life, not only on the basis of the genetic makeup, but using some sort of aptitude test, you can expect a surprise. The guys nearest to the society's ideal will always have stuff happening to them. Starting from small misfortunes to the possibility of being found dead just before the test.
Unless it is the real goal of your system, that is going to be a major side effect. With a huge part of your male population infertile/asexual, women have to jump through a lot of hoops to get pregnant. I even fail to imagine how this should be structured. Are alphas in effect male prostitutes, having no other job but to have sex? Or is it done as a medical procedure or artificial insemination? Even now in the western society a lot of women choose not to get pregnant, because pregnancy, childbirth and child care are major impositions. You are additionally complicating the 'getting pregnant' part on top of that.
So, in effect, such a society would be hard to create, hard to support, and even if you managed it, you'd get a rapidly ageing, genetically stagnating society, ruled by corrupt 'doctors'.