Dr. Verhaeghe is, according to most criteria, a successful academic, close to the top of the academic achievement scale and pretty high in the general social structure. To avoid ad hominem criticism, I will assume that Dr. Verhaeghe is an outlier, that he climbed to the top despite the goodness of his heart, and not because of some psychopathic personality traits on his part.
Having this out of the way, I can critique the logic of his argument on its own merit. Dr. Verhaeghe claims that
A highly skilled individual who puts parenting before their career comes in for criticism. A person with a good job who turns down a promotion to invest more time in other things is seen as crazy – unless those other things ensure success.
From this, he concludes that this system rewards career and penalizes one's love for his family. There are at least two problems with this conclusion. First, we don't know how other social systems perform in this regard. Did feudalism favour "success" to a lesser extent than the system Dr. Verhaeghe is critiquing? How about communism? Were there fewer psychopaths at the top of the social structures in the communist/socialist Yugoslavia or the USSR than in the current system?
I don't have quantitative answers to these questions, but neither did Dr. Verhaeghe offer any. I do know however, that the Yugoslav dictator Tito sent about 16 thousand political prisoners to something that looked more like a concentration camp than a prison. No one of the top Yugoslavian political or economic officials complained strongly enough to change this system. Did they exhibit more or less psychopathic tendencies than the people at the top of today's social structures in neoliberal societies?
Second, Dr. Verhaeghe's conclusion assumes that rewards and punishments are objectively determined outside of our minds. This is problematic because, from what we know about the logic of choice, human choice is based on subjective valuations. This means that the definition of success and failure is subjective. Each individual defines her own success. For me, for example, choosing a promotion over spending enough time with my family would be a failure, not success. I value time spent with my family more than a promotion if that promotion implies less family time.
Similarly, I value my family's happiness more than avoiding some stranger thinking I am crazy for choosing family over a promotion. All of us are free to make such choices because the value of alternatives is determined by our own valuing minds, not by the objective properties of the alternatives.
If we followed Dr. Verhaeghe's logic and assumed that the system is causing people to focus on their career rather than on their family, then we also assume that we have no power to change the system. How can I decide that I don't want to be a part of the neoliberal society if my valuation of everything, including that society is not determined by me? If career advancement always represents a reward and more family time always represents a punishment, then what if living in a neoliberal society always represents a reward? Then, even if Dr. Verhaeghe is right, there is nothing we can do about it.
 This is an unfortunate and often abused term, but let's stick with it without getting into a deeper discussion of its frequent use and misuse.