How to access "Faster (fine)" and "Slower (fine)" playback controls on VLC for OS X? Else, what alternative?
Yes, VLC for OS X provides this level of control.

If you go into VLC » Preferences [aka. ⌘+,] » Hotkeys you'll find that you can associate hotkeys for both actions – Faster (Fine) and Slower (Fine) – and more: 



You can find plenty of options here: 


set them up the way you were used to using them on Windows, 
set a hotkey for resetting playback speed back to normal etc.


Note: If your newly assigned hotkeys don't seem to be taken into account, you might need to quit VLC and start it again.