I used to have this issue too. I solved it temporarily by turning the Sound Enhancer feature off. I wasn't happy about having to listen to my music as "non-enhanced," but that was better than the deafening beeps.
I upgraded to macOS Monterey yesterday, and I'm happy to share that now, even with the Sound Enhancer turned on while listening to songs streamed in Lossless format, my Music app makes no sound beeps or any weird "stuttering" noises at the beginning of any songs. At least in my experience, it seems like Monterey on my 2019 16" MacBook Pro fixed these issues.
Good luck, everyone!