Why I reject the round-earth model is because I do not accept their evidence as accurate. Why I support the flat-earth model is that I do find their evidence compelling.
I appreciate pongo for making his clear statement. The obvious follow-up is "what evidence is not accurate", and "what evidence is compelling, and why?"
At the risk of this becoming a discussion, which it should, I'd like to explore these questions here. If the moderator(s) prefer, I will (or they can) move this to the previous discussion thread, or a new one. I hope that the stated policy of "no discussion in Q&A" will be reconsidered since any controversial "A" to a "Q" is likely to be challenged, or at least probed, which
should lead to a discussion. Since this is The Flat Earth Society forum, almost any answer is going to be considered controversial by someone. It's your forum, though...
Here are a few pieces of evidence I'd like your thoughts on, pongo (or anyone else, for that matter)...
First the Mainstream stuff:
There's an agreed-on globe that is consistent with measured distances and observed travel times anywhere on the surface.
The overall shape of the earth is very close to (but not exactly) an ellipsoid. The reference ellipsoid is close enough to a perfect sphere that in many cases the spherical model suffices, with the advantage of being significantly simpler. As equipment and techniques used to measure the size and shape of the earth's surface improve, the models (dimensions and orientation of the ellipsoid) are occasionally adjusted, but for the last few decades, these adjustments are small. For very high precision work, particular ellipsoid models that fit certain areas better than the worldwide model (at the cost of being worse elsewhere) may sometimes be used; the North American Datum of 1983 (NAD83) vs. the more general World Geodetic Survey of 1984 (WGS84) is an example. Sometimes older models are retained simply to avoid having to adjust maps when there is not a compelling need. This does not indicate "disagreement" among geodesists over the shape of the earth, at least in general; it is simple convenience or necessity in some cases.
Apparent motion of the sun and other celestial bodies across the sky.
The rise, path across the sky, and set of celestial objects is for most purposes exactly as you would expect to see for very distant objects from the surface of a rotating sphere with light traveling in a straight line. This includes the apparent motion of circumpolar objects (those that never set) around either celestial pole, and why some objects never rise at all from particular latitudes - true also of the sun at high enough latitude depending on the season. Exactly which celestial objects will be visible from any location and their position in the sky are routinely predicted with high accuracy based on the spherical model; they're there when you look, which confirms the predictions.
Small differences from the ideal due mostly to atmospheric refraction (about a half degree for objects very close to the horizon; quickly insignificant higher in the sky for most work) and, to a much lesser degree, the ellipsoidal (not spherical) earth, are well understood, explained, and predicted. In fact, the effects of refraction are observable with only a little effort: view or photograph the sun with a properly-filtered (and be careful!!!) telescope when it is high in the sky and note its shape is a very nearly perfect circle. Repeat a moment before it starts to go below the horizon. It looks distinctly flattened because the lower limb is refracted upward more than the upper limb.
The horizon.
Standing on the shore and looking out over large bodies of water, one can see distant objects disappear from the bottom up as distance increases. The same object can be viewed at farther distance before it's partially or entirely obscured by the horizon when viewed from higher above water level. Even if the surface of the ocean may not appear strongly curved on casual observation while standing on a beach, it is, very subtly. These observations, and the distances involved, are entirely consistent with a spherical earth approximately 8,000 miles in diameter.
Flat Earth:
There is not an agreed-on map of what a flat earth would look like.
I've seen on this site two types of flat-earth maps, the so-called "unipolar" and "bipolar" maps. They are wildly different from each other, and both representations show vast inconsistencies with distances and travel times that are known. Unipolar seems to be more popular here, but many will aver that they're only symbolic, not real maps.
Mapping should be much easier on a flat earth because only plane trigonometry is required, not the more complex spherical (or worse, ellipsoidal) trig. This is where it seems the appeal for the simplicity of a flat earth would be best demonstrated, yet there are no maps of even regions based on a flat earth presented. None. Even though making a true map of a flat earth should be simpler than what is routinely done, there's none at all.
Celestial motion.
The motion of the sun relative to the flat earth has to be in direct conflict with everyday observations without very peculiar behavior of light. The most obvious inconsistency is sunrises and sunsets, given the most-often cited model, with the sun circling daily somewhere near the equator, around the north pole of a unipolar earth. Unless the light is bending strongly, sunrises and sunsets simply aren't possible for geometric reasons. If you do accept that light bends, the reason for this has not been explained and doesn't appear to be understood, nor is the actual path a ray of light takes from sun to earth known. The bending of light would have to make objects in the sky appear lower than they actually are the further they are from the zenith, the opposite effect of refraction described earlier, so the sun should appear stretched as it descends toward the horizon. Has this ever been measured or even observed?
How does circumpolar motion of stars (and the sun) work south of the equator? On the Ross Ice Shelf during the southern summer the sun can be seen moving from right to left as you look at it, from high in the northern sky at local noon to low in the south 12 hours later, then back, once each day. As you travel further south, the same phenomenon continues, but with less difference between the highest and lowest elevations. I've been there and seen it. If I'm south of the sun, on a flat earth, how can the sun appear south of me unless light is taking a really complex path? Where has the simplicity gone?
The horizon.
"Look out your window. It looks flat to me." This seems to be the strongest direct flat-earth argument I've heard so far. Nonetheless, what you see "out your window" (presuming it's a view over a wide area with nearly-constant elevation) is not inconsistent with a view of the same from a large sphere. Inconclusive at best.
Objects disappearing from the bottom up, and, in fact, having a horizon at all on a flat earth again requires curved light of an unknown nature. If you have to presume that light can't be trusted to travel in a straight line, how can the Bedford Level Experiment be meaningful - if it actually showed what some people claim it did, which is far from certain?