I would think that the issue your program is running into is that having the ^s send another ^s inside of itself is creating an infinite recursive loop in which nothing is ever able to run past the place you invoke ^s. To prevent this, we can use the $ modifier as so:
$<^s::
SendInput ^s
return
From the relevant section of the Modifier section of the docs:
This is usually only necessary if the script uses the Send command to
send the keys that comprise the hotkey itself, which might otherwise
cause it to trigger itself. The $ prefix forces the keyboard hook to
be used to implement this hotkey, which as a side-effect prevents the
Send command from triggering it. The $ prefix is equivalent to having
specified #UseHook somewhere above the definition of this hotkey.
Edit: it seems to work fine for me even if I remove the $ modifier. Testing the following code shows me there appears to be no problems regarding code execution before, after, or during the SendInput statement.
<^s::
MsgBox no
SendInput, ^s
MsgBox yes
return
Maybe check your version or installation of AHK?