There are really only three storylines that the prequels would be needed for: 1.) the rise of the Empire; 2.) the fall of Anakin; and 3.) the circumstances of Luke & Leia being born, and being hidden. TPM has little to offer on those story points. It's AOTC and ROTS that actually flesh that stuff out.
I don't think anyone would've objected to starting the prequels off with a Chancellor Palpatine, and going from there. We didn't need to see kid Anakin in order to make sense of AOTC's teenage Anakin in Jedi training. Even Padme gets introduced thoroughly enough as a Senator in need of protection. Outside of Anakin calling her a Queen at one point, I don't remember her TPM role ever even being addressed in the next two movies.
As for explaining the Luke and Leia element, I will never stop regretting that Lucas didn't follow his own script (at least his ROTJ one):
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That explanation from his script actually made it into the ROTJ novelization, so many of us grew up believing it. I never read the novel myself, but older kids who did would relay the info to the rest of us.
And it works better on so many levels. Not the least of which is the fact that changing things in the actual PT means that Darth Vader may have wanted further revenge for what happened to his mother. His next target would be those Lars in-laws of his who failed to protect her. If he went after them (a reasonable suspicion), Luke was deliberately put in those crosshairs.
Plus, Anakin wouldn't need to be from Tatooine - which works well for many reasons too.