It’s all in the question. This noise is utterly annoying, and honestly pointless. It’s surprising there is no (easy-to-find) way to disable it in settings. Some sites suggest using third party apps, but I’d like to avoid that solution because I’m afraid it would drain the battery (plus, I shouldn’t have to do that).
If you use the default camera then
Tap on the ˇ on top of the screen
Now tap on the 
Disable the Camera sounds
If you use the app Open Camera then
Tap on on top of your screen on the right.
Now tap … More camera controls …
Disable Shutter sound
I don’t think the settings of a camera app have any effect on the system’s screenshot sound - except if you literally take a photo of your screen that is. 
You are right.
A complicated way to disable the screenshot sound is to set the Ring and notification volume to none.
But that is not practical to do every time and set it back after taking a screenshot.
As AndreasChris says, my problem is when taking screenshots, not pictures, so I don’t think it is going to help.
Regarding the Ring and notification volume, I would be ok with that solution, but unfortunately it doesn’t work. I tried disabling all sounds and setting all volumes to 0, the screenshot sounds remain.
But thank you for trying anyway.
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Disabling Ring and notification volume worked on my Fairphone.
Same here. Muting the phone works on my FP4 as well, but then the phone won’t ring or make any other notification sounds either. Do you use a custom ROM? What android version are you running?