There's what I consider 99.9% evidence in that we can see the big bang. Trying to argue that the big bang didn't happen to most astrophysicists is like trying to argue that the sun and moon don't exist. I see this big glowing thing in the sky, and I see the this big round thing at night.
Similarly, I see this big glowing thing in microwaves that in all directions.
There is a reason for that. If you extrapolate to t=0, then the "laws of physics" don't work, so you have to make up something weird.
We can't see them at all. If you look at 13 billion light years, you see the big bang. You can't see any further back than the big bang. We know the universe is finite in time, because we can see the end of the universe so to speak.
It's also not a technology issue. If you do galaxy counts, you'll find that the number of galaxies start going down well, well before the detection limits.
The current models don't say what happens at t=0. The backward guess ends at about t=10^(-52) seconds.