I think there are plenty of Democrats who do not want (or feel conflicted about) equal treatment under the law for adopted people, in regards to free access to our birth records.
The rationale generally given is a concern for birthparents' privacy rights. This privacy amounts to a special right that is not given to any other type of parent. In answer to that concern, I call your attention to the fact that birthparents who don't want contact are free to pursue legal actions (such as restraining orders) which are open to any other adult citizen who does not wish contact with another adult citizen. There is no moral or ethical reason for restricting the civil rights of adopted people solely because they might be someone's dirty little secret.
We all have freedom of association. We should all have the freedom to know where we came from.