I think the issue is an App's Auto-Correct settings take priority over [global] System Preferences. Those System Preferences will be in effect if the App has no separate settings, but if the app has its own settings they will be used.
Testing with Pages for example,
if the System Preferences are set to not check spelling:
System Preferences > Keyboard > Text > [ ] Correct spelling automatically (unchecked)
but Pages is set to check spelling:
Pages > Preferences > Auto-Correction > [ √ ] Correct spelling automatically (checked)
Pages will check spelling, overriding System Preferences.
Similarly, the Safari spelling settings will take priority over System Preferences
[Safari] Edit > Spelling and Grammar [ √ ] Check Spelling While Typing and [ √ ] Correct Spelling Automatically
This is similar to the scope of variables in programming. Local variables have priority over global variables.
So there appears to be no global system setting that will disable spell checking across the board.
ivan