It doesn’t “ruin the story” at all.
We see when Tarkin brings up Alderaan that the aim is adjusted so the planet’s in view, meaning the view is pointing towards the void of space by the time Leia walks in the room. The initial shot showing the planet is before she walks in.
Even if it were in view the whole time, “they’re gonna destroy it” is an odd conclusion to immediately come to.
Not to mention that’s such an odd thing to say about something so minor.