Any material from an Earth-Theia collision would settle into predictable orbits shortly (in geologic time) after the collision. Presumably, this is well understood by scientists and they would pick material that is not from that predictable band.
Understand that space, despite the gigatons and gigatons of material out there, is not an unpredictable or chaotic place. It isn't just ping-ponging all over the place, messing people up. If you obtain material from a comet, asteroid, or even a particular type of meteor, you can model where it came from pretty well by determining its orbit, as well as materials analysis of its components. There are just some rocks out there that could not have formed on Earth or on the planetoid that may have produced the Moon.
As for the evidence, the Big Bang is still the current mainstream theory as far as I know, and I do watch carefully for any significant changes to major cosmological theories. I'd like for one of them to succeed, because then things would get interesting, but while there have always been contenders, they're nowhere near as accepted. Don't grab on to the fact that there exist alternatives... there's always people who are working on them in good faith, but good faith does not mean that they are any more correct or accepted than the crank theories out there.
Don't get me wrong, there's more to find out there, but as time marches on, the discrepancies are getting smaller and smaller in well studied fields. The belief in a 6,000 year old universe was the product of Biblical interpretation, not scientific investigation and everyone knew it. It wasn't science making a mistake, it was simply NOT science at all. And Christians didn't necessarily believe it either, it had been posited by ancient and medieval scholars that the Earth was old and that the Bible timeline was not going to be precise. Arguing that there must be a bigger universe because we've found certain things is the same sort of argument as saying that the Speed of Light is not real because we broke the sound barrier, so we just need a faster spaceship. That's not how it works.
We know that the Webb telescope will let us see more, but the we have already have calculated the extent of the non observable universe from principles. If space-time meets certain properties and has certain values, we can extrapolate the size of the universe from existing data in the same way that I can give you a reasonably accurate circumference of the Earth by measuring shadows at noon on two relatively close places on the planet and applying certain calculations. I don't need to circumnavigate the Earth to do so, I only need to go far enough away to make up for the imprecision of my instruments. This was done accurately even in Ancient times by only having to measure in Greece and Egypt, a trivial distance. It can be done for the whole universe in the same manner through theories and observation of stellar bodies.