George Lucas did an excellent job of hiding the truth of Vader being Luke’s father from people, down to filming the scene one way and then bringing a elect few back to do the scene again. So, if it is as you say and the idea was abandoned partway, it’s very plausible the number of people who knew the epic twist would be a scant few.
Given the size of Naboo and the fact the droids go only after the Naboo and not the Gungans, it is extremely unlikely Binks would just happen to be at the landing site of any federation ships, much less the ship carrying Qui Gon Gin. He can’t sense droids maybe, but I bet Master Qui Gon was a big bright beacon.
When Qui Gon sees him, those ships aren’t moving fast and all things around are fleeing. Jar Jar makes no attempt to flee, but instead waves his hands and blocks QG’s path, then clings to him. Afterward, he follows QG, pledges to be his servant and when told “that won’t be necessary” replies “oh, but it is, it’s demanded by the gods it is.”
His whole initial dialogue with the two jedi focuses on two points, A, that QG has saved him once, then again, and he owes him and must be his servant. B, the safest place is Gungan City, but Jar Jar will be in great danger and trouble if they go, setting himself up to go with them when they leave.
During the dialogue with Boss Nass, they cut to Jar Jar as Nass is talking, and Jar Jar gesticulates in his handcuffs, perhaps steering nass with the force.
QG “overdoes it” when he uses the force to calm Jar Jar when the third fish shows up. This conveniently gives Jar Jar an excuse to close his eyes and focus on taking control of the nearest bigger fish to come rescue all three of them. He wakes up and clears his head, dramatically shaking it off the very moment the fish 4 takes its first bite of fish 3 and the dnager to them has passed.
Honestly, I half suspected you’d somehow tweaked or altered the balcony scene as part of an elaborate hoax, but having just watched it twice Jar Jar is just as stealthy, then suddenly becomes a clutz in time to present a target and then jump away and land on the opposite side, complete with droid firing high then snapping its head to Jar Jar’s position.
On the way up the ramp to the ship during the escape scene, just prior to your hyperdrive sabotage theory moment, Jar Jar gestures, and then is the second person up the ramp following Amidala/her double, with her honor guard in tow. There is now way an un-jedi influenced personal queen’s guard is going to let a Gungan they just met, a race that hates Naboo, be between them and their queen, right?
The first thing Jar Jar says on Tattooine is, this sun is doing murder to meesa skin. He’s aquatic. Why the hell is he going with QG and R2 to the city to find parts? QG seems appalled at the idea of taking Padme, so why is he cool with taking the clumsy/bumbling Jar Jar? Perhaps because Jar Jar contrived to be there.
When Padme is at her most uncertain, it’s Jar Jar’s mention of the Gungan’s grand army that gives her the push to go back to Naboo, and coincidently make Jar Jar critical to her plan, as he is the only Gungan she knows.