No one is saying that the experts are infallible. However, they have demonstrated strong evidence, using actual manuscripts, to prove their points. The documents and the history support their claims.
You are referring to "the experts" as though there exists a scholarly consensus that the NT was cobbled together centuries after the fact, and is riddled with errors and contradictions. This is a misrepresentation of the state of affairs.
As already mentioned, having spent a good deal of my life involved in biblical studies, including teaching on the college level, I have seen that - far from any kind of consensus - there exists a sharp divide between those scholars who approach their discipline from a position of faith and respect for the full authority of the sacred texts, and those who are dismissive of what they deride as "literalistic fundamentalism" and are dedicated to "proving" that the Bible is not an inspired document.
As for numbers, given the large number of conservative colleges and universities (and thus teaching positions in biblical studies), its a fair guess that there are far more scholars who accept the full authority of the biblical text than oppose it. However, the "liberal/progressive" faction controls important "citadels of power," including Ivy league schools like Harvard, Yale, etc, and of course they can count on support from the MSM (also ruled by anti-faith writers and presenters) should any issue spill over into the news cycle.
However, this does not mean that all positions are equally valid - or invalid. As the old X-Files tagline put it: "The truth is out there." My experience in the field has thoroughly confirmed my faith in the full authority of the biblical texts. I believe if anyone is willing to question the "Received Truths" of the skeptics who reject the very idea of sacred scripture, there are sufficient answers to any troubling questions.
This may involve, for instance, looking beyond the first several Google hits which are typically controlled by the "powers that be." The real eye-opening evidence is often on the 2nd and following pages.
Of course, the theological skeptics/liberals are fond of pointing out that their opponents approach the subject from a position of faith. Fair enough. What the skeptics/liberals fail to admit is that they approach the topic with every bit as much faith as those they oppose. No scholar is capable of personally investigating every fact or truth-claim presented by any scholar.
Scholars are of necessity very narrowly-focused, and are forced to either accept or reject the claims of other scholars in specific fields that are beyond their area of expertise. A professor specializing in one area of Old Testament studies is not in a position to personally evaluate the claims of, say, an expert in Near Eastern Studies with specialty in Assyriology.
What happens in practice is said professor is already likely seeking support in these other areas for a position which he or she is already committed to. So it is not a matter of being "objective" but rather mustering support for one's worldview.
I do not mean to present a pessimistic view of scholarship, as I firmly believe that approaching biblical studies from a strong position of faith will often lead to surprising evidence that the prevailing skeptical culture has ignored or suppressed. Put another way, one cannot "go wrong" by humbling oneself to the divine authority of sacred Scripture and desiring to uphold those texts.
I for one would rather face the ridicule of the enemies of God's Word lest I fall afoul of the words of Jesus: "If anyone causes one of these little ones, those who believe in me, to stumble it would be better for them to have a large millstone hung around their neck and to be drowned in the depths of the sea."