That's not a problem with TCustomControl itself. It does work, I checked.
The problem is probably missing Canvas functions. You need to make a list of all canvas functions TStringGrid needs and see where they are implemented. They are usually implemented on LCLIntf as WinAPI functions, like Rectangle, LineTo, MoveTo, ExtTextOut, etc, etc, etc.
There are also some non-winapi drawing functions on LCLIntf such as Draw 3D rectangle.