Not really. No one have Hashirama's SM and i'm pretty sure Kabuto/Orochimaru gave Madara that Hashirama face on his chest. He wouldn't have RSM unless the sage gave him his senjutsu chakra. If Sasuke never got Hagoromo's chakra i'm pretty sure he would have awakened a standard Rinnegan from Hashirama's cell's/chakra considering Hagoromo said Hashirama still have some Ashura chakra...
But the Hashi face comes from having Hashirama cells, and we didn't see one on Kabuto so it further re-enforces the idea he didn't have the 1st Hokage's power.
If Sasuke had cells implanted in him like Madara we would have seen a face on his body, and he wouldn't need time to adapt to his new eye since he would have had a boost in chakra reserves.
But you're right he wouldn't have awakened RSM, but it's not because the Sage didn't give him Senjutsu. Hagoromo gave both of them the same chakra (Indra + Ashura) but because of their different abilities they awakened different feats.
Sasuke had Indra's chakra and MS which allowed his eyes to evolve while Naruto had the chakra of 9 bijuu and Ashura's body which allowed his BSM to progress to Six Path Senjutsu.
Regarding Sasuke's rinnegan, he didn't awaken a standard one because he had Hagoromo's chakra added to Indra's as opposed to just having Ashura's chakra added to Indra's. Basically, Indra + (Indra +Ashura) = tomoe Rinnegan while Indra + Ashura = standard Rinnegan