That's why the biggest change was cleverly altering and reediting the footage so that, instead of hanging in space like a fat machine moon with nothing to do, the Death Star was actively attacking the Rebel base on Yavin IV. Yep, that was all added later. You ever wonder why the Death Star didn't just shoot the planet Yavin, which was in the way, and destroy the moon that way? Because none of that stuff was originally filmed. In the original script, they weren't under a direct threat.
The whole notion of the Rebel base being on the brink of obliteration by the Death Star's giant laser (thus putting a ticking clock on the mission) was created in post-production with whatever they could cobble together. That's why you never see any onscreen character mention the base being under attack, and instead we're given offscreen voiceovers and random cut-ins of computer screens saying "The Death Star is totally about to fire on us, you guys!" That was all their attempt to add excitement to the sequence.
You can see what the original would have looked like here, or read the shooting script. The whole third act is just lifeless. The motivation for the final sequence would have boiled down to "The Empire is probably going to destroy our base at some point in the future, so we'd better take out the Death Star as a preventative measure." Important side note: Film editors don't get nearly enough credit in general.