Thanks for the answers!
I think, I will not install the "Screen Recorder" app because first, it does not record sound (but that is what I need in the first place) and second, because it needs "Full System Access" rights to be installed. I think, access to the screen, to the sound and to the user's files in the user's directory should be enough. I do not see, why full system access is needed.
In the comments of the Screen Recorder app i saw the comment "works great, thank you, no longer have to record my remote desktop". Maybe this is a solution for me. Does anybody know how to connect to a UT phone via remote desktop?
(Another solution, which would be fine for my use-case, would be to simply record only the sound. Is this possible in some way?)