What you're describing is fixed point. Fixed point absolutely exists in practice, but it has an important limitation: its accuracy is scale-dependent, unlike floating-point.
Suppose for example you were doing a simulation of the solar system. What units do you use for coordinates--AUs, light-seconds, kilometers, meters, even miles? With floating-point, it doesn't matter: you get the same relative accuracy at the end. With fixed point, you get vastly different relative accuracy if your numbers are ~10¯⁶ versus ~10⁶.
Suppose for example you were doing a simulation of the solar system. What units do you use for coordinates--AUs, light-seconds, kilometers, meters, even miles? With floating-point, it doesn't matter: you get the same relative accuracy at the end. With fixed point, you get vastly different relative accuracy if your numbers are ~10¯⁶ versus ~10⁶.