Hand shields bluetooth antenna in Google Pixel 4a 5G?
If you're having troubles with your new Google Pixel 4a 5G from late 2020 in 2021 or later I recommend downloading the latest Google connectivity apps and update your system to  the latest Android version.
I was forced to set-up my device offline, the original version must have had some kind of bug with the bluetooth. After installing the updates everything worked.
I was able to determine, that the issue is software related after conducting a few tests. It turns out, that not holding the phone but turning on its screen is causing the stuttering.