Hell, my first ever car rental (to go from DC to Harrisburg because that leg of my flight got cancelled) was a walk-in with Hertz, and the slowest part was picking out a car.
Re: getting to the car in the first place, just about every airport I've been to has a bunch of rental agencies onsite (either walkable or via a shuttle bus), and I suspect HSR stations would be no different.
It's a security feature. It's more secure for them to retype everything then to try and verify that everything you typed matches the ID you put in front of them.
When you have status they've already done all that once and they trust you because you've successfully rented from them before. Without status, they have no reason to trust you, so they have to verify everything.
It's a risk/reward trade off for them.