Cramer's random model of the primes allows the prediction of features of the distribution of the primes even though the model is not about "real" primes but about randomly generated "quasi-primes" (unlike the real primes, such quasi-primes can be even, for example, or there can be two consecutive "primes"). Terry Tao writes that due to these models, mathematicians are in "the curious position of being able to confidently predict the answer to a large proportion of the open problems in the subject [prime number distribution], whilst not possessing a clear way forward to rigorously confirm these answers!" (Source: https://terrytao.wordpress.com/2015/01/04/254a-supplement-4-probabilistic-models-and-heuristics-for-the-primes-optional/)
The Cramer model has thus drawn the attention of philosophers like James Robert Brown and William D'Alessandro, who argues that such models "can help us understand a phenomenon even when it offers no explanation of the phenomenon." (Source: http://philsci-archive.pitt.edu/21225/1/Unrealistic%20models%20final.pdf)
My question is whether there are any other purely mathematical examples of cases where inferences can be drawn about a mathematical phenomenon from models that actually represent not the phenomenon in question but a completely unrealistic/idealized/distorted "quasi-version" of the phenomenon?
Thank you very much in advance.