Naruto and Sasuke were both able to meet the sage because they both were able to achieve half of the samara
(9 bijuu=RM)
(EMS+senju DNA=rinnegan)
Upon meeting the sage they were gifted with halves of Hagoromo's power. Which enhanced their half of the samsara.
Naruto RM Kurama mode (yang increase gave him better reflexes and physical ability than normal RM) (might have made him a better Bijuu container as well)
Sasuke Yin Rinnegan (Sharrinnegan (Kaguya's eye) works with genjutsu Sasuke's eye is similar, but weaker, it's a rinnegan that is closer to Kaguya's eye, but still a rinnegan. It carries the power of the Rinnegan and Sharingan
The sage specifically said that when Ashura, Indra and Kurama's chakra mix together, that's when he can appear. Sasuke and Naruto fused their chakra avatars together when they fought Obito. Only when they almost died did they go to the deepest parts of their own subconscious to meet Hagoromo.
I agree that they didn't get their power ups from the sage. Naruto gained his from combining the bijuu chakra, because just like a Juubi jin gathering all of the bijuu's chakra, they can awaken Rikudo mode. The sage has nothing to do with gaining that power.
Sasuke did get Ashura's chakra when Naruto gave his own chakra to sasuke and Hashirama also did the same. Dna has nothing to do with it. It's specifically the chakra that doies the job. Sasuke could've awakened the rinnegan with any of their chakra, after almost dying once. kabuto only healed sasuke, nothing else.
The sage gave them Naruto the sun symbol, which is yang and Sasuke the moon symbol, which is yin. It was never shown that the sage gave them anything else other than half of the chibaku tensei seals.