We study a class of finite-state channels, known as POST channels, in which the previous channel output serves as the current state. A POST channel is deemed approximately memoryless when the state-dependent transition matrices are sufficiently close to one another. For this family of channels, under a surjectivity condition on the associated memoryless reference channel, we show that the feedback capacity coincides with the non-feedback capacity. Consequently, for almost all approximately memoryless POST channels whose input alphabet size is no smaller than the output alphabet size, feedback provides no capacity gain. This result extends Shannon's classical theorem on discrete memoryless channels and demonstrates that the phenomenon holds well beyond the strictly memoryless case.