The excuse for the ship's travel to Tatooine is a complete fiction. Actually, it was required to give the queen time to explain the situation to the Jedi and, in case of a negative reaction, to contact the Hutts. The Jedi, faithful to the Council's order to protect the queen, didn't consider the drug dealing a grave crime and agreed to continue overseeing the queen's protection. Nevertheless, the malfunctions which were discovered in the ship's drive after the jump to Tatooine played an important part in Anakin-Vader's life, bringing the Jedi into Watto's store.
The scene of Anakin meeting the Jedi and Amidala is quite accurate, as well as the conversation between Watto and Qui-Gon Jinn. There is some inaccuracy in describing Watto's behavior. According to the recollections of a person who knew him back then, Watto (quite in accordance with Tatooine customs) was ready to accept republican dataries, but only in cash and with a 40 per cent discount. Qui-Gon insisted on a cashless settlement (because it was difficult and quite costly to convert Republic currency to cash on Tatooine) and 20 per cent discount at the most.
Of course, Amidala could turn to the Hutts, but their unpredictable attitude presented a certain risk, since the Hutts could easily decide that it would be better for them to sell Amidala to the TF. Luckily for the three, Anakin, who found the strangers likable, pointed at Watto's weakness – gambling.
With this information at hand, Qui-Gon started an argument with Watto about whether or not a completely unprepared person can participate in the Boonta Eve Classic, after which they agreed that Watto rents a pod and gives it to Qui-Gon on the security of one of Amidala's handmaidens (the story of Amidala following Qui-Gon disguised is accurate except for a single detail – both Jedi knew who was the queen of Naboo).
If Qui-Gon's participant will make it to the finish line alive (whatever his time result is), Watto was to sell the hyperdrive cashless in republican dataries with a 25 discount. If the person wins, Watto agreed to transfer to Qui-Gon the ownership of Anakin (who by that time already has demonstrated a force talent). As the participant Qui-Gon proposed Obi-Wan and thus the outcome of the race was sealed.
The popular opinion about Anakin-Vader's participation (and even victory) in the race is nothing but a myth, which was invented by the sect of Vaderworshippers (which is all but destroyed by now) in the times of Empire's heyday. It must be said that the opinion about Anakin-Vader being the most powerful Jedi in the galaxy and the mistaken opinion that he was the one who turned the scales during the battle of Naboo are also purely Vaderworshippers' made.
Something that must be noted separately is the theory which was voiced in the film, about the connection between a Jedi's talent and the presence in his cells of a virus like form called "midi-chlorians". Sadly, the ignorance of the sensations-loving masses forces the makers of films to connive at the most lowly tastes, for example using beautiful, yet long since rejected theories. By the time of the events described in the film it was already known that the so called "midi-chlorians" are nothing more than parasitic, Force sensitive forms of life which can indeed be found in a Jedi's body but cannot be the source of his abilities simply because there were many reliably recorded cases where no MC's at all were found in a Jedi's body.
Anyway, the fact of Anakin being Force sensitive was registered by both Jedi unambiguously, and the task of getting the boy to Coruscant and organizing proper training became of primary importance to them.