Why would I run a rail-to-rail input opamp with its inputs at the rail even if higher voltage supplies are available
The AD8628 is a 5 volt op-amp and has a maximum supply rail of 6 volts hence why it can't run from +/- 5 volts. They use it here because of its phenomenally low input offset voltage of typically 1 uV. It also has an input voltage range that includes both power rails. It sounds like a good choice to me.
Andy already answered for this specific case, but in general running an op-amp with wider rails than you need to can be problematic.
There are a number of electrical characteristics which are dependent on the supply voltage which may or may not be an issue with a particular application.
By limiting the rail you also limit the extent of the output. If you never want the output to be positive... using +5V for the top rail would be a bad idea.
As you indicated, using wider rails means more power lost. Although that may be a small change in quiescent current, if your load is normally being held at 0.1V with 20mA of current, on a 15V supply, the op-amp will need to dissipate 0.3W.