Hagoromo didn't give Sasuke an eye. He gave Sasuke half of his chakra. Why are people still talking about Sasuke awakening his Rinnegan from Senju DNA? That was already proven false when Hagoromo told Naruto that the Rinnegan awakens from having Hagoromo's chakra.
Madara combined Ashura and Indra's chakra to get the Rinnegan. Sasuke received chakra from Hagoromo and got the Rinnegan.
I agree that Hagoromo didn't give Sasuke the eye and only gave him half of his chakra. However, I don't believe that that is the reason that Sasuke awakened his rinnegan.
It has been explained that Hagoromo's chakra in an Uchiha with EMS produces the rinn'egan, but I think Sasuke had the chakra necessary to awaken the rinn'egan without Hagoromo giving him anything.
I think that RSM and The Rinnegan are ultimately equivalent to one another and that they are ultimately different ways of producing Hagoromo's chakra, and one or the other + being a incarnation of Indra and Ashura is necessary to meet Hagoromo and gain his gifts, both can be obtained without meeting him.
So In my mind it goes like this.
For Madara. EMS/Indra chakra + Ashura Chakra + death = Hagoromo's Chakra = Rinnegan: judged unworthy so no meeting or gift.
For Sasuke. EMS/Indra chakra + Ashura Chakra + death = Hagoromo's chakra = Rinnegan: judged worthy so meeting and gift/moon mark.
For Naruto SM/Ashura chakra +9bijuu Chakra + death = Hagoromo's chakra = RSM: judged worthy so meeting and gift/sun mark.