Is God's word truthful in all that it tells us?
We can be certain about what it teaches us about God and the nature of our relationship with Him, which is the important part concerning us personally.
So when "scholars" say "probably not..." I think we can safely say that we would rather believe what the Bible says, than take scholars' opinions as fact. Just because some scholar claims a contradiction, it doesn't mean they are authoritative. After all, Jesus was born more than 2000 years ago. What can they possibly know about that?
If there are any apparent contradictions in the Bible concerning minor things like details of situations, number of people, and such, how could that possibly destroy our faith in God, if we know Him and have a personal relationship with Him? The Bible is literature, which includes poetry, songs, narratives, testimonies, symbolism, prophecies, and all kinds of writings, that when people search for inconsistencies, they find them, according to their assessment. But why should that affect our faith in the only true God and Jesus Christ whom He sent?
So then, it depends on what is meant by "all that it tells us." If that means every single statement in it the way those statements are interpreted in modern times, then I think it is setting us up for failure. If the ramification is that: if it fails the "test" then we can't trust any of it, then I think that it's putting God to the test. It depends on what "truthful" is meant - if it means by someone's particular standard, then that someone may find out in the end that one should not judge God or the word He has provided, as opposed to letting God's word judge us.
In that video, the idea that Jesus was not born in Bethlehem is based on one "scholar's" writing, and the statement "most scholars" is merely an assertion. The Bible is clear that Jesus was born in Bethlehem, so we should believe what the Bible says, not what some "scholar" reasons out in his fleshly mind.