Is that all it takes to get unemployment now? Just an application and they take your word for it? I thought you needed to get legitimately laid off (i.e. not quit or be fired for cause) by an employer in the state who notifies the state of this fact for you, and perhaps who has to pay part of your benefits.
And there's always both upsides as well as downsides to handling things federally versus state.
You wouldn't need to handle it centrally, you could do something like log all "single-state" transactions against a name, SSN, & bank account, and publish to reach of the other states your data -- preferably through a federal data store, but it could still work otherwise?
For certain things, this perhaps one of them, there are benefits if handled on a federal level with more oversight.