100000... is only equal to 011111... if your digits roll over, but if your digits roll over then you have a p-adic number system (specifically a 2-adic system), and the exact same argument would hold equally well for the 10-adic applied to 10000..... = 99999.....

You argument isn't dependent on being binary or not. Only that the numbers roll over.

In bases higher than 2, you can avoid the issue entirely by saying that some digits get +1 and other digits get -1.

If the generated number only has digits 1-8, then 0000... vs 9999... is obviously not a problem.