The fine structure constant is usually defined using $e$, $h$ ,$c$ ... However, from QED, we know it cannot be derived but only experimentally measured. Does that mean the usual definition we use in terms of quantities we know is not precise or it simply says that same as the fine structure constant, $e$, $h$ ,$c$ values can only be experimentally measured.
It is the fine structure constant we measure most often because it is a dimensionless quantity, i.e. its values does not depend on the unit system used.
Indeed, that the fine structure constant is an experimental input to our theory means that, equivalently, we could also consider Planck's length, the elementary charge or other dimensionful constants as the inputs, yet it is more convenient to combine these into dimensionless constants whenever possible.