If we had a single state-run bank (even if it were just as the fulfillment "back end" behind different consumer-facing servicing organizations), then a transfer would occur on a single ledger in a matter of milliseconds, rather than waiting for the East Podunk Bank, Trust, and Pork Cannery to pull a batched update file over a T-1 they put in in the first Bush administration.

Fraud could be handled better than it is now. We have terrible security/convenience compromises (Chip + Signature, anyone?) and when we try to improve matters (3-D Secure) it is presented so badly and clumsily that it basically dies on arrival (it's coming back, but mostly because the EU is cramming it down people's throats)

Chargebacks are a weird case. It feels like they come from two places-- either as the first response to a fraudulent transaction, or as an escalation for when there's no other way to get satisfaction. If you're fair with your refund policy, your chargeback rate should be close to nil.