Or, Qui-Gon and Obi-Wan could be on a mission to take the world leaders (Padme) originally to Coruscant, while searching for any leads as to sith activity - and leads as to why they have revealed themselves. The Tatooine part would be unavoidable, but it would be Obi-Wan who finds Anakin - however, Qui-Gon still sees greatness in him (no prophecy here!) and takes him with them. As for the senate discussions, they should only be mentioned at most or totally wiped out - it slows the movie down, and it's just one of those moments.
Just after they leave to return to Naboo, the Sith launch a full-scale attack on the Republic and seize power. When Qui-Gon and Obi-Wan find this out, they hide. Meanwhile, they explore Anakin's abilities.
Yoda? I don't know, I haven't come up with something.
Palapatine? A known Sith leader, NOT a Naboo senator.
Well, I guess it sounded good when I first thought of it - but I can't tell anymore. It may just sound like a bad fanfiction by now, and if so, just give your thoughs on the very beginning (Sith assault).