Viewing in light of the whole trilogy and not just within the confines of ANH, I always kinda assumed it went down like this:

Han was enroute to Tatooine with a shipment of spice for Jabba. The Star Destroyers that were in orbit (the same ones that chase them as they leave Tatooine) or some smaller ships dispatched from said Star Destroyers intercept and board the Falcon as it approaches the planet. Han obviously dumps the cargo at some point prior to being boarded. He passes the inspection and lands safely in Mos Eisley. He’s got no money thanks to failing his delivery so he’s now looking for a way to get off planet ASAP. Which explains why he’s hanging out “where most of the best pilots can be found”. He needs a job and quick! Which also explains why he takes such obviously risky passengers.

In the meantime, Jabba gets wind of Han’s failure and immediately puts out a bounty on him and people like Greedo start combing the most likely places to find him.

So, in typical “Han” fashion, it’s not some long drawn out drama. All of that happened VERY recently and it was a whirlwind, fly-by-the-seat-of-your-pants adventure that led him to even consider taking the job of carrying Luke and Co. which then led to the next wild adventure rescuing the princess and so on, etc. etc.

ALSO: it’s worth considering the deleted scene that had Jabba living on Tatooine even in ANH. While it may not have been in the final cut of the film, as far as George was concerned Jabba was located on Tatooine long before ROTJ made it canon. Heck, it was in the comics (and novel?) back then. Which means it HAD to be something along the lines of my ideas above all along.