No, they can't. A parallel argument against writing the Bill Of Rights was that it was completely redundant: every enumerated right was already protected by the limited granting of powers to the government. Remember, the US government starts as nothing and is granted highly restricted powers (not the imposition of limits on an otherwise unlimited government, as most seem to think). There are plenty of other rights, regardless of enumeration, which likewise the government is not authorized to limit.

The problem is there's an overlooked difference between positive rights vs negative rights. Positive rights, enacted by legislation, require others to facilitate one's actions. Negative rights, requiring no legislation save for active protection thereof, require nothing more from others than benign neglect. The whole premise of the US Constitution is full recognition of negative rights: lawmakers are not empowered to hinder anyone's actions beyond a small and specific delegation of powers, much less compel others to facilitate someone's alleged positive right.

The "create new rights based on their personal conceptions" you worry about are wholly positive rights, anathema to the Constitution and core American principles because they are based on compulsion of others to facilitate that new legislated/adjudicated "right". Rights, as understood in the Constitution, are nothing more than people doing what they see fit so long as others are not wronged in the process, and the legislative & judicial processes exist solely to identify & resolve these conflicts when they do arise.