Federated learning (FL) has emerged as a promising distributed learning paradigm for training deep neural networks (DNNs) at the wireless edge, but its performance can be severely hindered by unreliable wireless transmission and inherent data heterogeneity among clients. Existing solutions primarily address these challenges by incorporating wireless resource optimization strategies, often focusing on uplink resource allocation across clients under the assumption of homogeneous client-server network standards. However, these approaches overlooked the fact that mobile clients may connect to the server via diverse network standards (e.g., 4G, 5G, Wi-Fi) with customized configurations, limiting the flexibility of server-side modifications and restricting applicability in real-world commercial networks. This paper presents a novel theoretical analysis about how transmission failures in unreliable networks distort the effective label distributions of local samples, causing deviations from the global data distribution and introducing convergence bias in FL. Our analysis reveals that a carefully designed client selection strategy can mitigate biases induced by network unreliability and data heterogeneity. Motivated by this insight, we propose FedCote, a client selection approach that optimizes client selection probabilities without relying on wireless resource scheduling. Experimental results demonstrate the robustness of FedCote in DNN-based classification tasks under unreliable networks with frequent transmission failures.