McEliece-like cryptosystems, which are at the core of code-based cryptography, exploit that efficient decoding of a suitable known code is easy but decoding in a random code is hard. The secret key is a fast decoder for a secret code, whereas the public key contains a disguised representation of the chosen code. Thus, it is crucial to ensure that the inherent structure of the secret code can be well hidden from the attacker when algebraic codes are used. We aim at building McEliece-like cryptosystems based on linearized Reed-Solomon (LRS) codes and their decoding properties in the sum-rank metric. We thus investigate their rich structure and appropriate disguising strategies. Since LRS codes generalize Reed-Solomon codes in the Hamming metric and Gabidulin codes in the rank metric, we show how known distinguishers for these code families carry over. In particular, we discuss the generalization of the square-code and Overbeck's attack. Surprisingly, LRS codes are not fully vulnerable to these attacks, which keeps them interesting for further cryptanalytic research and/or the design of cryptosystems.