This is not possible unless:
- You have root access (delete the sound file), or
- Use a 3rd-party trusted screenshot app, or
- Disable system sound (yeah, it will disable notifications sounds and all too)
I'd recommend you to use a 3rd-party app (you can use power and volume button to take screenshots by the app by over-riding system's screenshot system). You can try Screenshot Assistant from F-Droid, or Screenshot or Screenshot sharer (super helpful if you don't use Google Assistant much) from F-Droid. FYI, F-Droid is super trusted.
The Silence option refers to the notification you see with the preview of the screenshot. Try editing some other options in there (try making it alerting and then enable 'Pop on screen'). You will see what I mean. And yes, you're correct in saying that setting it as Silence or Alerting doesn't actually make a difference to the sound - it doesn't occur anyway.
There is no user facing control to silence the sound emitted while the screenshot is being taken.
Source - Pixel Phone Help - How to turn off screenshot noise