Install a clipboard manager.
For example, I use clipmenu. I start it whenever X is started by putting this
CM_OWN_CLIPBOARD=1 clipmenud &
in my ~/.xinitrc.
Yours is a common misunderstanding when talking about X selections.
From Arch Wiki: Clipboard:
According to the selection protocols, nothing is copied until it is pasted. For example, ...
In normal mode:
s same as cl
S same as cc
[count]s same as [count]cl or c[count]l
[count]S same as [count]cc
In visual mode:
s same as c
S is remapped by vim-surround to a (very useful) surround operator.
I don't see any clear advantage of having s for substitute when there is already a c for change.
I'm testing with nmap s ys in my init.vim (or .vimrc) ...