As for the Falcon, it was busted up in Empire to physically represent the sacrifices that Han had made in the years since he hooked up with the Rebellion. His fortunes had clearly fallen since the end of ANH, thus the Falcon is in a poor state of repair. Chewie starts to repair the Falcon just as Han resolves to finally leave and go pay off his debt. But the Empire attacks, interrupting Han's independent plan and Chewie's repair efforts. Likewise, the Falcon's flight is always interrupted by something. If the Falcon were in perfect working order, it would subtly clash with the disorder of Han's life. And people would wonder why a smuggler with a perfectly good starship hadn't used it to pay his bills.
Even in its disordered state, the Falcon was a formidable prey for Death Squadron. Just like Han, the morally disordered "scoundrel," was a formidable ally for the Rebels.