Lunarians are only immortal in the sense that they're perfectly healthy, both physically and spiritually, meaning that they never grow old or sick. They can almost certainly tank more damage than a normal human, but they're not indestructible.
Kaguya is a Lunarian, but she gained a completely different kind of immortality from drinking the Hourai Elixir. Her body can be killed, but her soul cannot, and it will restore her body to full health whenever things go too badly wrong (even if it has to create a new body from thin air).
Also, most of the time when a Touhou character acts like they're going to kill someone, they're just talking big.
For humans the physical body is the core of their being; mental/spiritual stuff is important but secondary (i.e. a human can survive depression, but not decapitation). Youkai are the opposite - their spiritual side is what defines them, and their physical bodies are secondary (albeit the exact ratio varies from species to species). Asserting your identity is to most youkai what food and water are to humans, and vice versa. Under the Spell Card rules, talking big is one of the few ways youkai have left to be youkai, so almost all of them do it. The human characters do it because... when in Rome, do as the Gensokyans do.
