Would it though? The agencies are at different levels - local vs. federal. In addition, you are notifying the post office you have moved, not the government. Your federal notification is basically to the IRS, and not everyone pays income tax, so integration there would not work. Also, not all taxpayers are eligible to vote - another complication.

You have oversimplified the details of this sort of integration and conflated local vs federal institutions. The best way to do this change would be through your state's BMV/DMV, and many states already update your registration (or offer to) when you submit an address change there.

Registration and voting are easy and well documented in every state in which I've lived or researched.