I've said this before in discussions here about this situation, but I was born out of wedlock and put up for adoption. My birth parents came looking for me, decades after the fact, and I was not very happy about it. I actually refused to meet my birth father at all. Everyone is different, and not everyone feels a longing to know biological relatives when they are happy in the life they have. This case, however, is very, very different than that of ordinary people. The cat was out of the bag on Delphine's parentage once the biography of Queen Paola came out. Once there had been such widespread publicity, I can understand why Delphine wanted to be acknowledged (perhaps even more so than before her story had been exposed to the world at large). As things stand, she has the worst of both worlds. I think King Albert's refusal to do so made the situation much worse. As Johan says, King Albert must have some strong private reason to have behaved as he did, such as a promise to his wife or children, but -- unless there is some huge surprise hidden in the DNA results -- it will not have done him or his family any good in the end.
I assume none of you has ever had to deal with being an out of wedlock and unacknowledged child. Psychology has answered why children want to be acknowledged a long time ago. Delphine de BoŽl has been trying to be acknowledged over a decade, but the book just gave her the opportunity, since this was widely known, to be open about it. As I said, it is a matter of human psychology.
As for the law, the EU has forced all member states to change their laws on out of wedlock children, that are now considered equal to the so-called legitimate ones. They have equal rights to property, which is also why Albert of Monaco also had to acknowledge his illegitimate children, beyond the trials, and assure them of one fourth of his estate after his death. I am aware of other EU countries that were, for example, very secretive about adoptions that were all closed. The EU forced them to change the law because EU law supersedes national law. The rationale behind EU law is that it is against their human rights for children not to know their parents and/or not to be acknowledged and be made heirs.
When Albert II was king, this law was not enforceable because, according to the Belgian constitution, the king is above the law. However, now he is no longer reigning king, so the law applies. If the Belgian court hadn't decided to force him to undergo this DNA test, Delphine de BoŽl could have taken this very easily to the Court of the European Union, where she would have won. Belgium would be fined in this case, and the law would have to be enforced. She also has the option, after that, to go to the European Court of Human Rights, which will also find in her favor.
I am aware of several legal cases in EU countries outside Belgium, where children found such hurdles. Once the EU ordered the countries to change legislation pronto, there was a wave of children getting the information they required and rights.
Think of it a little like the obligation to support a woman one gets pregnant. One has the option not to get someone pregnant. If he does, then, that's that. The child has rights.
In the case of Albert, I truly think he is just very stubborn, something members of the dynasty have the reputation of being. Legally and morally it is unfair that he does not acknowledge his daughter, of course. And it really is no big deal. As other members have observed, Albert of Monaco acknowledged his two illegitimate children, albeit after some pressure.