I'm sure that not every rabbi holds of the 6,000 year thing. Because after all, there are many Orthodox rabbis who do not believe the buba maiseh that the world is only 6,000 years old (there is not a single source for that). So it's only natural that that time is also not the "end of the world" Or whatever. That 6,000 years thing can be metaphor or mashul. It is aggada afterall. But anyway, hopefully redemption will come even sooner than that date. If it doesn't come we keep trying, that's all. Only unreasonable/illogical people will lose hope if it doesn't come by a certain date.
But there is certainly plenty of sources for this in the Talmud.
Please do not make vague statements.
I said that there are no sources that the world is 6000 odd years old (5770 plus 6 days). Because there is no source for that. Period. Every opinion is that 5770 is the years from the creation of MAN, not the world.
As to the talmud, I already addressed that. The 6000 years world, 2000 years this, 2000 years, and 2000 years moschiach, etc - It's an agada. It does not mean someone has to say the world is literally 5770 years old because of that, and literally no one does. Until maybe the current generation or a few decades ago, NO ONE said that. End of story.