It is a big problem.
The problem is two fold. History by it's very nature is not science. There is hypothesis and experimentation. It is what most probably happened and it's possible impact on events afterwards. The second is the time period. It is not unusual at all from what it seems that mythacysts like to make out.
1. Acid free paper that can withstand time was not invented. So little informationi s not proof! It just means we do not have much records besides copies of copies of copies
2. The culture at the time was pre-enlightenment where people were not taught to investigate and use logic and critical thinking to figure the world around them. It was pissing contests and egos both with philosphers who tried to be intellectual and people praying to different Gods and using religion
3. The Gutenberg Printing press was not invented yet. This means you had to hand write and copy something if you liked it including all the errors and additions of the copy of the copy of the copy of the copy until the editors had several different versions of something and had to guess which one to use when copying by hand
4. Very high illiteracy of around 97%. This number is much higher in rural areas like Galee where Jesus would have preached
5. Paul most surely existed as Christianity was spread and all the leaders of the ancient church cite him as he spread it through Athens by the Greek philosophers. Sources all point back from the late 1st and 2nd centuries
6. Christianity was small and considered Jewish for the 1st century.
7. We have Josphious and Tactitus.
So I am into this subject but not religious anymore. So far scholars have looked at it this way:
1. Book of Mark probably written about 55 - 65 AD near Rome or in Rome based on writting style and events (Jesus said ressurection would happen in disciples lifetime and no mention of the destruction of the temple in 70 AD yet). Jesus is not God and became raised to diven status after death (the later paragraph was forged where he resurrected due to writting style differences and grammar)
2.: Q (this is a document used to create Luke and Mathew as it was obviously copied as too much looks verbatim. We do not know name or have surving copies) 70 - 85 AD. Mathew and Luke used Q and Mathew used Mark. Jesus now was made devine at his baptism. THe texts were corrupted as one version in 120 AD quoted paslm 29 where Jesus was made devine at the baptism to something that just re-affirms he already is devine.
3. Book of John he always was God and was creater of world circa 95 AD to 100 AD
So yes the NT is problematic as it is biased but historians use the information out of it by looking for similarities. So no smoking gun but it is not like we have 0 evidence at all.