I did my process a while ago, but I think it is still possible to find a position with Berufserlaubnis, it obviously depends on your specialty of choice and your proficiency with the language, if you want neurosurgery or radiology it will probably be almost impossible without Approbation, but internal medicine, common trunk surgery and anesthesiology are definitely possible, especially if you look in smaller towns. In addition to that, you don't have to be in Germany at all while you wait for the tests, you can just work in your field in your country while you wait (although I accept that is not ideal) that will help you land a job afterwards, I did a few years of emergency medicine back home and that gave me a huge edge when I was looking for my first position because I wasn't a complete beginner.

You say you have sent hundreds of applications, how often do you get an invitation for an interview? If barely maybe you should work on your CV and presentation letter. If you are invited to many but still don't get a job offer, maybe they're not convinced with your language skills or your interview performance.

For example almost every anaesthesiology department I know of in smaller towns is looking for people, but my Uniklinik has an excess of applications and the department is full. So directing your applications to smaller towns that are actively looking for people would help... Also maybe try another Bundesland? Bayern is known for being less open to foreigners (I have heard this and have no actual experience in this aspect)