Err, the law requiring social media to register also has very broad provisions for government mandated removal of content from these social media. So it's not exactly unlikely that that would have been the next step after the registration deadline had passed.
You are completely right that social media needs some form of regulation to avoid their nasty sides btw. But that's not what this law was seemingly intent on doing.
It's a bit more complicated than that. This specific SS boss had already been assassinated by Dutch resistance in 1943.
What likely happened is that after the Jewish owner of the portrait died in May 1940, his collection was placed in possession of Göring. He then sold many of the works in October 1940, and presumably this is when Seyffardt purchased it (we don't know for certain). After he died in 1943, the painting presumably passed through his son and ended up with his granddaughter.
The case only came to light now because another family member discovered that he was family of Seyffardt, which he didn't know because the Seyffardts changed their name to avoid association with the Nazi collaborator.
So since there was no knowledge or record that this painting was ever in possession of Seyffardt, and he had already been busy being dead for 2 years when the war ended, the trail just ran cold.