Alderaan had 2 billion people. The support for the rebellion there would be amount to what? Thousands? Hundreds of thousands? A whole founder planet which was a showcase of progress wiped out because Tarkin was a moron. If they were keen to solve the situation, they just needed to depose the Organas after the Senate was dissoluted and put some loyalist or moff in charge.
>The plan drastically backfired, driving thousands of beings to the Alliance cause, even in the Core Worlds, and sending many planets in the Mid Rim into open rebellion. A number of Alderaanians serving in the Imperial military defected to the Rebel Alliance after learning of their homeworld's destruction; among the defectors was the TIE fighter pilot Tycho Celchu. The Imperials attempted to pin the blame for Alderaan's destruction on the Rebels, keeping a small number of people loyal to the Empire; however, the ruse convinced few.
Alderaanians were influential in the politics and military since forever. It was like one of those soviet purges that fucked up the whole military by killing some of the best officers around.