Differential and Linear Cryptanalysis on Random Sbox
Typical attacks on Secret SPNs (i.e. with secret uniform S-Boxes) are integrals, see e.g. [1].
Even for AES: the S-Box is too good against differential/linear attacks (strong mixing in MixColumns is also necessary), so best attacks are of another kinds. (Not counting related-key attacks, which allow to deactivate many S-Boxes).
More to the question: in some special cases it may be indeed useful to consider differential cryptanalysis with ``random'' difference. See for example [2], where the output of the Feistel function is very small so that it does not matter which difference to choose - the differential will be sufficiently strong anyway.
[1] Biryukov, A., Khovratovich, D., &amp; Perrin, L. (2017). Multiset-Algebraic Cryptanalysis of Reduced Kuznyechik, Khazad, and secret SPNs. IACR Transactions on Symmetric Cryptology, 2016(2), 226-247. https://doi.org/10.13154/tosc.v2016.i2.226-247
[2] Orr Dunkelman, Abhishek Kumar, Eran Lambooij, Somitra Kumar Sanadhya:
Cryptanalysis of Feistel-Based Format-Preserving Encryption. IACR Cryptol. ePrint Arch. 2020: 1311 https://eprint.iacr.org/2020/1311