Mullahs already aren't supposed to be performing the nikaah without proof of a state marriage; doing so can result in a fine. This was largely put into place to protect women's property rights since a state marriage is necessary to legally protect her. This is very important.
But in practice, that law doesn't seem to have made much of a difference. It's easy to simply have the groom or a family member pay a fee to the mullah that's equal to the fine, in case there is a problem for the mullah. It's also not always practical to get a state marriage because in rural areas, the state offices aren't open often, or couples may have to travel a long distance to get the marriage done. Since the nikaah is a community-sanctioned form of marriage, then why should we expect couples to bother with the state marriage just because of a fine?
The article quotes Munara Beknazarova saying:
Munara Beknazarova of Open Line, a Bishkek-based NGO that offers support to bride-kidnapping victims, said many village mullahs are aware that abducting a bride is “against Islamic principles,” but still bless marriages if the bride says she has consented to the union. “By the time the mullah arrives, [the bride] has often been physically intimidated, occasionally raped, and threatened with social exclusion,” said Beknazarova. “Of course she consents.”Yes. But honestly, how would this law solve that problem? It's not like getting a state marriage means that the marriage was completely consensual. It just means they took a little more time to get the paperwork done. Kidnappings by their very nature are hasty affairs, whether they're consensual or not. Having the nikaah done first does not necessarily mean the marriage shouldn't have happened.
The law would help solve this problem though (although, as I mentioned above, it's easy to get around):
Beknazarova maintained that an unregistered union denies a woman, and, ultimately, her children, of her civil rights because she has no legal right to alimony or protection if she leaves an unofficial marriage. This helps cement the practice of bride kidnapping as normal in rural Kyrgyzstan.
I do think it's not unreasonable that more laws are passed trying to curtail bride kidnapping, but NGOs should be aware they are probably going to be largely symbolic. I think laws can contribute to the solution, but that's not where the major changes in bride kidnapping is going to happen.