No, you're not missing anything. Identities are meant to be meaningless internal-only surrogates which are for the computer to allocate and deal with, and the computer doesn't care whether you use positive or negative numbers.
However.. it's never quite that simple. There will always be a human somewhere trying to read these things, comprehend them, and reconcile them. We just find it harder to think about massive negative numbers than about massive positive ones. Don't make it harder on people trying to make your system work. Eventually these will leak - into reports, external references, screens. Try telling customer minus two billion they have to use a phone key pad to enter their customer ID to access their account!
If you think there is any chance of overflowing the identity in any likely future just use the larger datatype straight away. The considerations that drove Y2K problems are past. Disk is cheap. The extra memory used per query can be balanced against the peace of mind knowing the application is designed for a 50 year lifespan. You won't have to implement extra monitoring against the day 4 (or 8!) years from now when the last integer is allocated.
I know of one system which overflowed and the identities were decremented by 2.1Bn, effectively re-starting at int.Min. I've seen another use negative identities but fail because logging cast them to varchar(10) truncating the minus sign. I know another where the identity was defined as numeric(18,0), just to be sure. And I've seen another overflow without a plan in place, bringing the system down for some time. Because when you do reach int.max you have, by definition, four billion rows to deal with, and that doesn't make for a fun weekend.