Sorry that this is a semantic view, I know this will come down to definitions - but I would like to work on a logical and practical basis, so if making reference to specifics, please show the actual real world use, rather than theory.
Judaism is a term often used interchangeably to mean a few things, broadly a religion, an ethnicity, a race, and an ethno-religion.
Of these definitions and understandings, religion is the one that seems to actually "work" by which I mean when you look at Jewish institutions the entry criteria is whatever degree of adherence to the Jewish faith. There are not tests for ethnicity, or race, for entry (perhaps there might be somewhere in the world, my view is somewhat specific to England), to a Jewish school, or membership to a Jewish organisation, or synagogue.
As such, on a practical level, in day to day life, if you are born to Jewish parents, but at a young age convert to Islam, you would not be admitted to a British-Jewish school by merit of birth alone, you would need to personally believe in the Jewish faith. (an example of this happening might help to change my view)
And again, on a practical level, this makes perfect sense. You can test for religious belief with a conversation, and verification from a religious figure that they attend prayers however often, that they haven't been observed to break any of the fundamental laws and so on.
What would an ethnicity test look like? And for a racial premise would it even be accepted if a school started doing genetic testing as part of its entry criteria? Such schools in England are called faith schools, not ethnicity schools.
This also makes the ethno-religious label somewhat redundant, as again at what point is the ethnicity angle ever actually practically useful?
Hopefully some of my questions here can be answered, and if any aspect of my belief is unclear, please ask clarifying questions so I can help keep us on topic!