Actually, that's not the case.
The ancient Greeks (starting at least with Pythagoras, 6 centuries before Christ) had worked out the earth had to be a sphere (using scientific principles), by the time of Pliny (1st century AD) the notion was generally accepted, including by most of the church fathers, such as Augustine, and many later clerics (such as the Venerable Bede).
The only real doubters were a minority of early Christian authors (St. Cyril of Jerusalem, St. John Chrysostom, Diodorus of Tarsus and a few others) arguing for a 'flat earth' on purely Scriptural grounds. The Church occasionally produced further proclamations in favour of a 'flat earth' (such as Bishop Vergilius), but by the 8th century AD, the matter was more or less settled, at least among the learned.--and probably the general populace as well. Our ancestors may have had less education than we boast, but they were surely far more observant of the changes of the seasons and the events of the night sky; and you can't make much sense out of even casual astronomy assuming a 'flat earth.'
It's a modern myth that prior to Columbus folks thought the earth was flat. That was a poetic myth created by Washington Irving, who wrote a popular book portraying Colombus in heroic light. Now, of course, we know that Colombus was pretty unsavoury, but that is beside the point.
Now, none of the above takes anything away from the intent (as I read it) of your first point in your posting: scientific 'truth' is not determined by opinion polls. It is determined by doing science.
Credit for a convincing argument that the earth is a sphere is generally given to Aristotle. He used the usual arguments (observations of mariners, etc.) plus the fact that the earth's shadow on the moon during lunar eclipses is always circular, and only a sphere could always cast a circular shadow.
A generation later, Eratosthenes provided a strikingly accurate calculation for the earth's size, by measuring the shadow of two objects at noon, which were a known distance apart (the figure of 500 stadia is often given). The sun was directly overhead for one object (a well which was lit up right to the bottom) and about 7 degrees off-center for the other, so every 7 degrees was 500 stadia, and for 360 degrees ... he had the earth's circumference. (His writings are lost, and there's another way he might have done it, but this post is long enough.)
This was around 250BC, and all educated people from then on knew not only the shape but also the size of the earth. Columbus, however, used a different calculation, and didn't realize how far he had to sail to get to China. But he knew the earth was a sphere.
As you say, the true "science" of a matter isn't determined by opinion. Likewise, true "science" generally has a provable theorem, generally mathematic, that can be proven from start-to-finish in some way. "Sciences" like anthropology, palentology, sociology (thrown in for all of the "social scientists in this world"), et. al. are made from observations and selected findings. Selected findings can many times be just an available set of bones that have been found. DNA tracing has been the closest thing to science, in my opinion to date, but still involves some bit of probability and chance (you'll never hear an expert on the stand saying "there's a 100% surety that so-and-so's DNA IS the same as exhibit A").
In the end, they are still observations and the theories are mainly the result of the influence of human experience on the thought process working overtime to establish "how it must have been."
I don't know the truth, and I don't think the truth can be conclusively proved (unless somebody has a time machine we don't know about). Then, what remains is belief, and that is what sets the two groups apart. One content in the spiritual, and the other using the argument as a means to control.