The configuration of op amp you have drawn would certainly be unable to properly operate at the most negative end of the power rails. This is why a lot of op amp applications use a proper negative rail hence 0 volts (ground) is somewhere close to the midpoint between supplies.
The circuit you have drawn would work with inputs close to (or slightly above) the positive rail of course but. if you want an op-amp that can operate with inputs across the whole supply range you need a far more complex input circuit.
If you look at this BJT op amp you can see the complexity: -
There are two diff input pairs - one formed by Q1 and Q2 and another formed by Q3 and Q4 - at somewhere mid supply one pair takes over from the other pair thus allowing the full input range to be resolved. Note that the circuit above is also rail-to-rail output too. The diagram is of an LT6220