To be honest, I do not know Neo4j's GraphML format. However, yEd was never intended to work with Neo4j in the way you would like to use yEd.
(GraphML specifies how to encode structural information. Applications are free to store any other kind of information like geometry data, visual data, meta data, etc. in whatever way they want. As a result, there is no "default interoperability" between two applications solely because both applications support reading and writing GraphML files.)
The recommended way to interact with a Neo4j database is indeed the data explorer for neo4j.
The data explorer is actually not a cloud-based solution, but a client-side browser application. I.e. it runs locally in the browser on your machine. Moreover, it does not send data anywhere. It connects to a Neo4j database, pulls the requested data into the browser process that runs locally on your machine, processes said data in said local browser process, and displays the corresponding diagram.
If you connect the data explorer to a Neo4j database that runs on your machine, your data never leaves your machine.