I do have two things I ant to add on this matter. If I don't my heart will pop out of my chest, sprout limbs and start beating the shit out of me. I just can't end this without getting this off my chest.
I have two theories on why the whole 'Eirin is (not) immortal' business.
1) Eirin is indeed immortal, and has consumed the Hourai Elixer. However, she has consumed
a flawed Hourai Elixer.Think about it. Mokou used to have black hair, and didn't have any magic power whatsoever. However, once she drank the Hourai Elixer, her hair turned white over the years, and she obtained the power of the Phoenix. Kaguya still has black hair, and she doesn't have phoenix powers She fights with her own magical knowledge. Kaguya got an elixer from Eirin, genius of the Moon. Mokou stole hers from some bloke on Mount Fuji. No one can make the Hourai Elixer as perfect as Eirin, who had Kaguya's eternity powers to help her out, and the Hourai Elixer is nigh impossible to create for mosts. This dude had made/obtained an imperfect Hourai Elixer, which, while giving immortality, doesn't protect one from diseases, and makes your hair age none the less. This gave Mokou white/grey hair: her hair didn't stay youthful.
The Hourai Elixer needs an 'eternal ingredient'. Kaguya could make a perfect one with Eirin because she used her power of Eternity. The Fuji man didn't have an eternal princess around, so he used an alternative: the soul/feathers of a phoenix, an eternal being. This however also gives the drinker of this Elixer phoenix powers. Thus Kaguya stayed the same as she is, while Mokou got uber buffed to pyro extraordinair.
My point here?
Eirin also has white hair, like Mokou, while she claims she doesn't age. I think Eirin tested a proto type of the Hourai Elixer on herself, one which wasn't prepared well or lacked the Eternity power provided by Kaguya. Thus, once Eirin reached Earth, her hair started to turn grey/white thanks to the impurity of Earth. Her hair aged too.
On the other hand, maybe Eirin only got hightened regeneration and eternal youth, but can still be killed. That's why Yuyuko didn't mention anything about Eirin being unable to die: she still has a few of those 'death points' Yuyuko uses.
Or alternatively:
2) ZUN ignored the 'immune to diseases' point to increase drama, or was drunk/forgetful while writing it. Everyone in Gensokyo is an unreliable narrator, even ZUN. It should just be ignored entirely, like how Perfect Memento musn't be taken for truth either. ZUN tends to fill in bullshit for trolling/due to his own lacking memory/'cuz he's drunk. And drinking that much beer all day will have negative effects on your memory capabilities...or your brain cells in general.