Many businesses are built on the premise of offering more customer centric versions of existing government services. For example, FedEx is intended to be faster and more dependable than USPS. Similarly, H&R Block has made a business of making it easier for people to submit their taxes online, which is a service also offered by the IRS. And, private security firms fill in for the police and military.
Wherever government sees the need to offer its services, there is typically high demand. Because private businesses can choose to ignore cumbersome clients, they're able to deliver on more ambitious customer promises. So, if a business is able to offer a better experience, there are always plenty of people willing to pay premium prices. Any service that you wish government offered more effectively / efficiently is a business opportunity waiting to be unlocked.