Sidious does not, to my knowledge make a statement to that effect. Yoda does, and Mace concurs, however. We might as well accept that as fact.
However, there is no evidence supporting that "bringing balance to the force" refers to eliminating the dark side completely and especially the Sith specifically. Sure, I guess I can concede that it's a possible meaning, though it doesn't sound like one. However, if that IS the meaning of the prophecy, then why would anyone be thinking about such a prophecy during a time when they don't think any are about anyway?
I don't think your interpretation holds up to to any sort of close or careful reading. I'm not sure I follow your middle paragraph. I don't think I can see logical patterns you believe are self evident. If there are sith and no one thinks there were any, then the most natural of responses is to investigate. But I can't see how they anyone, even Qui-Gon, would want to tout the fulfillment of a prophecy that, as far as they're concerned, is already fulfilled. IF the prophecy refers to the elimination of the dark side in general, that's a little more believable, but it does certainly mean that there are more weilders of the dark side than the single master/apprentice Sith.
However, the problems the Jedi are going through seems less related to the existence of Dark Side users than some other internal problem. They can't see as well as they should, their feelings are clouded, they don't know how to respond to things. This is not a problem of just the Dark Side being around, or else they'd NOTICE that they can't see something they should. If they're that blind, it seems to me it would be something they're doing wrong.
I never argued Qui-Gon though Anakin would make more Sith. I'm not sure wehre you got that. He doesn't know, they can't see the future, and they don't know how Anakin will bring balance to the force. I don't think that his initiation into the Sith is necessarily part of the fulfillment, it just happened due to poor choices, but in the end he finally straightens out.