yEd simply does not have the necessary command-line interface because you are not allowed to use yEd in an automated process like "exporting via script". Please see the yEd Software License Agreement, §1 License Conditions.
We can only provide yEd as a free service because we license the underlying programming libraries to customers. By allowing programmatic access to yEd, we would effectively tell our customers to not license our commercial library but use our free tool instead, which would quickly result in our company going out of business, taking yEd with it.
If you are interested in a programming API for generating high-quality diagrams and the possibility to "hook" diagram generation "into Doxygen", please take a look at the yFiles programming libraries.