Indeed there is a simple argument to see that:
Assume by contradiction that there is a (transient or steady) potential flow u=\nabla\psi, satisfying u\cdot n=0 around the obstacle, and u=(1,0) at the inlet. Denoting u=(u_1,u_2), since u satisfies \mathop{\rm div}u=0 and \mathop{\rm curl}u=0, it follows that the couple (u_1,-u_2) satisfy the Cauchy-Riemann equations. Hence u_1-iu_2 is holomorphic in the domain. But on the inlet we have u_1-iu_2=1. By the principle of holomorphic continuation (using the Schwarz reflection principle to extend the domain of definition beyond the inlet) it follows that u_1-iu_2=1 in the whole domain, hence u_1\equiv 1 and u_2\equiv 0.
But this contradicts the condition u\cdot n=0 around the obstacle.
Next another remark is that if you put only the condition u\cdot n=-1 at the inlet (and no condition on u_2), you loose the uniqueness. Indeed you have a steady potentiel solution (that satisfies u_2\not=0 on the inlet), and at least another steady flow with non-zero vorticity (that satisfies u_2=0 on the inlet).