OK, with Midi-Chlorians brought into the debate I feel I need to explain the importance of Qui-Gon.
To save time and to make it easy to discuss, I'll go in point form:
-We get introduced to Qui-Gon in TPM and quickly learn that he doesn't follow along with the mindset of the Jedi Order, who after a thousand generations have become complacent enough to dissect their faith.
-This complacency lead them to discover Midi-Chlorians, which to the Jedi scientists explain how they got
strength; how they can communicate with the force. In this discovery, faith was no longer the guiding, erm, force of a Jedi's life.
-The Jedi Order follow the what they call the Unifying Force, which is an overall view that seems to ignore the subtle hints that the Force gives out. Qui-Gon tells Obi-Wan to follow the Living Force, which is the pulse that binds all the elements together. Essentially, He's trying to teach Obi-Wan the same thing that Yoda was teaching Luke: Faith.
-The Jedi Order would never have acknowledged Anakin if not for Qui-Gon's insistence, and if Anakin never became a Jedi, he never would have given balance to the Force. Arguably, If Qui-Gon lived and defied the Council's orders, Anakin would have been raised learning the force the same way Luke ended up learning it, and he wouldn't have become jaded by the obvious complacency of the Jedi Order, and no doubt, he would have been protected from Sidious (at least until Sidious found a way to kill Qui-Gon) ((since its pretty clear by Sidious' story about Darth Plaegous that Anakin's inception was intentional and his discovery by the Jedi either followed his plans perfectly or severely altered his plans))
-Qui-gon's death is a direct result of the Order's complacency too. The council refused to believe his claims of the Sith Lord. If they followed the Living Force as Qui-Gon did they would have taken this threat seriously and sent a pile of Jedi.
-Let's assume that it is Qui-Gon's death that turns Count Dooku, his former master, away from the Jedi. Lets assume further, that Dooku actually intended to join the Sith to destroy them only to be corrupted as quickly as Anakin was.
-(not to take away from the above point
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Let's assume further, since the OT have some pretty serious power given to names and their importance -Anankin died, killed by Darth Vader, as an example- that Count Dooku was once a Jedi known as Sifo Dias, who was killed about ten years before AOTC shortly after the death of Qui-Gon. Since we know that Sifo Dias commissioned the the clone army and we know that Tyrannus hired Jango Fett, is it too far fetched to assume that both men are actually count Dooku? Think about this one for a while, and if you discard EU like Lucasfilm does, it holds more water than my Aunt Ginny's ankles.
-After Order 66, and maybe before then, Yoda is communicating with Qui-Gon, and pledges himself to the slain Jedi's training, in doing so, turning Yoda into the Jedi Master that trained Luke with a focus on the Living Force and not the Unifying Force. Faith. Not Midi-Chlorians. Even though for 800 years, Yoda trained Jedi the exact opposite.
-Qui-Gon then went on to train Obi-Wan the same thing.
-Bottom line, with out Qui-Gon's influence, the entire story would have unraveled differently. There is no telling what Sidious' plans for Anakin were before he was discouvered by Qui-Gon, but the dialogue in ROTS clearly indicates that Sidious had a hand in Anakin's creation.