Robbedo, Welcome!
Undead assassins is an interesting theory. I've seen sillier, I've posted sillier.
I'm going to use the terms undead, resurrected, reanimated, etc. but what I mean is people who were dead and are now walking around. Thoros might have said of Catelyn, "she's not only merely dead, she's really most sincerely dead." I don't mean people like Davos, Aeron, or Patchface who could have swallowed water, become unconscious and been revived. I do mean people who are morally, ethically, spiritually, physically, positively, absolutely, undeniably, and reliably dead!
We've only seen a few of the Faceless Men... the old man and Jaqen, I'm not including Arya's little friend since she seems to not have graduated the Faceless school yet. But two things leap out at me. First, Young Stormlord hit the nail on the head... if Jaqen was already dead then why would Jaquen be afraid to die? Second, we've seen undead before... Beric, Catelyn, Ser Waymar, Small Paul. Undead beings seem to carry their appearances in death back with them in life/unlife and I see how a Faceless Man could overcome this, but the mindsets of Beric and Catelyn, notably, at the time of their deaths became their only mindset when they were reanimated. Waymar and Small Paul were just blind killing machines when they were resurrected. But Jaqen and the old man seem to have total control over a wide range of emotions for undead.
And I think the quote is not "Only life and can pay for death", rather I think it's "Only death can pay for life." There's more than a subtle difference here.
It may be that the Faceless Men ceremonially "kill" their acolytes. They may do something like the Ironborn do or it may be even less severe. Thus the Faceless Man is now dead to the world.
If you've lurked around the forums or if you've read a fair bit of posts, then you'll know I'm not the end all, be all of knowledge on ASOIAF. I post like it, but I'm not. My posts are only my opinions of the moment. If you or someone posts an opposing view, I might just change my opinion... I've done it before.
Again... Welcome!