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The truth is the Lucas was tinkering with the script and story throughout the process. Destroying Alederan came late in production, after they shot the scenes of Luke and Leia talking after Obi-Wan died. That's why Leia doesn't seem concerned her whole world is gone.
Also, we don't know how long the DeathStar can go between shots. The video assumes it's a rapid fire weapon, but it looks like it might take minutes, if not hours to recharge.
It's probably based on the deathstar blowing ships up in ROTJ, but that would've been negligible power compared to blowing up a planet.
Another thing.
Leia could have been in communication with the Rebel base while they were en route.
Leia = I've got the plans, but I think this ship is being followed.
General = Come in. If they lose you they'll probably start blowing up planets at random to get us to attack. Most of our fighters can't go into hyperspace, so we'll be stronger if we fight them here.
Darth Vader: Hey, do you want to know my secret identity?