While it isn’t generic, it will work fine for what you want to do. Just edit it with the parts editor and change the pin labels to match the pins of your new IC and change the metadata to match the new IC save it as a new part and you should be away.
Essentially yes, the schematic will be generic but in many cases thats all you need and its easy (or at least easier). If you want internal logic or an odd pin layout you need to edit the underlying svg files to do that, but for something that matches the generic template just changing the labels does it. Fritzing doesn’t do simulation (at least directly) so it doesn’t really care what the chip does internally only what it looks like.