Ya I kinda messed up my explanation but looks like you caught on rather well anyways haha. I do know that he split his chakra into yin and yang and I agree with what you said about hagoromo's senjustu.I already know that. You divided the requisites into samsara, hermit mode, and hermit chakra, and i asked what the difference was between the latter two, but lets put that aside for a moment. First of all you do know that hagoromo split his own chakra into yin and yang right? so if u take sasukes yin and add it to narutos yang "minus tb" then you have hagoromos senjutsu, hermit mode
however as u said samsara is needed as well, so considering sasuke already has the samsara aka rinnegan, then giving him narutos chakra by your theory should awaken the third eye am i right?
i say possibly, because if this is the case why didnt the sage awaken the third eye? or are u saying that he did?
Regarding the sage not gaining the third eye, it's complicated haha. I do stick by what I said that if that happened to sasuke, he will probably be able to awaken the third eye but what you said about hagoromo is interesting and brought up this thought. What if hagoromo did have the requisites to unlock the third eye after becoming the jubi jin, gaining a power that was almost on par with kaguya's but instead, he didn't use it and he split his chakra up because he believed in the possibility of peace in the world?