Title. Although I could be wrong and it was just Madara hyping himself up, with regenerative powers like that it's highly likely there's an extended lifespan/ aging stops. Why do you think hagoromo died of natural causes then?
I'm going off memory here...but during the time Obito had the Bijuu extracted, i think Kurama said something about him being in a state of being unable to move and also mentioned that Gedou mazo had a large lifeforce which was still inside Obito...which was the reason a Juubi jin wouldn't die from extraction.(where else could Kurama gain this info except from Hagoromo.)
Now while it was said that hagoromo made the moon on his deathbed...It was also stated that extracting the gedou mazo from a juubi Jin would lead to their death...Ie Hagoromo would start dying once he moved the mazo from himself. But he is the ancestor of Uzumaki, so like kushina he was probably able to hang on for awhile due to his powerful life-force.
wasn't it due to Ashura that Hagoromo decided to make the Bijuu and not becuase of his age (Speaking of which, how old was hagoromo when he died huh? remember he should have had Uzumaki longevity so he might have been way older than he looked)
so, did he really die of natural causes?...or did he commit suicide by extraction.
As for Juubi Jins immortality...factors to remember hagoromo never befriended Juubi, so it could have been feeding off him every time he used it's power. It's power could have been damaging his body.
Would it be different for a perfect Jinchuuriki...Madara might actually have been considered as such, since Kaguya wanted him to absorb her...so she might have cooperated.
As for Madara saying guy almost killed him...how would he know his limit's after just becoming a jinchuuriki? Remember Naruto didn't even know he could fly.