9 Bijuu Solo - Differences

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Hagoromo didn't give Sasuke a Rinnegan either. He gave him the Rikudō no Yin no Chikara, the moon-seal and his chakra which is the source of the Rinmegan. Sasuke had to awake it after all. Madara on the other hand needed nearly his whole life to achieve the Rinnegan.

The Rinnegan can be attained only when the user attains both of the transmigrants chakra

Spoiler

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Technically Sasuke should've attained Rinnegan same as Hagoromo although he attained Rinnegan w/tomoe [which clearly has Hagoromos contribution in it as well - As you mentioned most likely the Yin chakra can be a catalyst for the mutation, although cannot be certain about it ]

P.S. It's true Sasuke attained Rinnegan at a young age unlike Madara [Maybe it's awakened when the person is near death - although which cannot be justified]​