This was so fricativing informative! I have been learning German on Duolingo for two years and have never felt comfortable with the way I pronounced "R", as I learned how to r-r-roll my tongue's tip but did not understand what goes on at the back of the throat. Never expected such a clear and pointed presentation to solve my problem. Kudos to your teacher too!
This is common. The trill occurs when you hit a sweet spot. So for a normal fricative, you want to avoid that sweet spot. That being said, the two sounds are "allophonic" in the language - meaning people perceive them more or less as being the same thing.
Why does it seems so hard? I can get the sound bu just for half a second and than it is gone.If I continue practicing, will the sound get better? Is there any other exercise to improve this "muscle"?
To me, it seems like that I necessarily need at least a little bit of saliva on the tongue to speak it properly. Is that a correct assumption for a native or am I "cheating"?
The saliva flailing about gives an acoustic quality to it, but it's not necessary. The essence of the sound is the air pushing through the narrow channel created by the palate and the tongue.
Another thing about the name Moritz is it has a tricky /ts/ sound at the end. The key to that is to place the tongue in the /t/ position early, cutting off the airflow of the /i/ prior, giving a brief pause of silence. When english speakers say it, they don't give that break in airflow, and that contributes to the accent, along with mispronunciation of the uvular fricative