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How to arrive at the coordinates for the sun from any position on earth?
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The video in the OP shows many vectors representing the two descriptive angles that position the sun in the sky.
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A vector includes a magnitude but we're not concerned with that for this exercise, so it can be ignored.
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All we're looking at is the DIRECTION of each vector, given by the compass bearing and vertical angle above the horizon of the sun.
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In astronomical observations ascension and declination are used for this purpose, by which any star in the night sky is located.

That’s nice and appears a slam dunk ... until you realise that the ability or inability to reconcile the vectors to all meet at a point would depend upon what projection of the globe model onto a flat surface one is using.

One could just reverse the process by assuming all the headings have to agree in that sense and then create a projection that places all of the origins of the vectors in relative positions that make them meet at a point.

That would be mathematically possible. Whether such a projection is feasible based on other data like known distances among locations and how light travels is another question (I believe it’s a priori impossible for ANY flat model to preserve all distances in all directions if a spherical model accounts for them)

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The accuмulation of all the data could turn into a considerable task, especially when one considers the fact that the geoid produces level lines that are not neat and clean aspects of a perfectly spherical earth.
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If I could find the video where this is explained very well, I'll post it.
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In other words, a so-called vertical line at point A might point to the star Vega, but then another so-called vertical line from point B only a mile away might point about 2 or 3 degrees away from Vega. 
This cannot be explained by the curvature of the earth (where a nautical mile translates to just one minute of arc, and 4 seconds of sun movement in its course), but it CAN be explained by the variation in level due to perturbations in the geoid.
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What is "level" at point A might not be "level" when sighted from point B, by some small amount.
And then you could have a point C even further away where a level line of sight confirms that of point A again.
The geoid has highs and lows something like the surface of calm water, which average out to the curvature of the earth.
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With the aid of satellites more and more refinement in the contours of the geoid are being developed over the years.
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Therefore, it is no longer a question of whether the earth is spherical or not, but just how irregular its shape is in approximation to spherical.
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That's why we say it is a SPHEROID and not a sphere.
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There is a place in eastern Canada where a major river would appear to flow uphill if you only consider its astronomical direction of flow.
That's because the attraction due to gravity changes direction along the river as though the center of the earth would appear to be moving as one follows along the path of the river.
But if you check with surveying instruments you find the water is flowing where "downhill" would appear to be based on how gravitational force is directed at each point.
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That’s nice and appears a slam dunk ... until you realise that the ability or inability to reconcile the vectors to all meet at a point would depend upon what projection of the globe model onto a flat surface one is using.

One could just reverse the process by assuming all the headings have to agree in that sense and then create a projection that places all of the origins of the vectors in relative positions that make them meet at a point.

That would be mathematically possible. Whether such a projection is feasible based on other data like known distances among locations and how light travels is another question (I believe it’s a priori impossible for ANY flat model to preserve all distances in all directions if a spherical model accounts for them)

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The projection being used is very similar to the one used in a standard star plotting tool used by navigators, when you look at the templates for the various ranges of latitude.
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Star Finder 2102-D originally developed by Captain G.T. Rude, shown at minute 3:20 in the following video has such templates:
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This star finder has a base table that images the celestial sphere but the templates shown are for latitude ranges of 10 degrees.
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If you watch this video, be aware the narrator mispronounces Greenwich as "Grin-witch" when it should be GREN-itch.
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But his descriptions of the various meanings of celestial navigation basics is quite good.
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He describes the Base Disk and the 9 latitude templates at minute 17:30.
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Stop the video right at 1:19.  Now clip the lines after the point at which they converge at the tip of the pyramid.  That's where the sun would be on a flat earth model.

Or, perhaps better, stop it at 1:46.  At the tip of the pyramid, before the lines pass through each other and diverge into space, that's where the sun would be.  There are a handful of extraneous lines around the edges, but that could actually be explained as reflection off the dome.

Stop the video right at 1:19.  Now clip the lines after the point at which they converge at the tip of the pyramid.  That's where the sun would be on a flat earth model.

Or, perhaps better, stop it at 1:46.  At the tip of the pyramid, before the lines pass through each other and diverge into space, that's where the sun would be.  There are a handful of extraneous lines around the edges, but that could actually be explained as reflection off the dome.

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Okay, let's take the "better stop" at 1:46.
Stopping the video there, I count 16 vectors in your "pyramid" which is actually closer to a cone, than a pyramid. On the basis of those exceptions to the rule, those 16 vectors, you are suggesting that we should IGNORE the majority of the vectors that point entirely elsewhere, of which I can count at least 64. The latter divided by the former is 4:1.
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There are FOUR TIMES as many vectors that point AWAY from the quasi-convergence that you seem to think is so important.
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Why would you want to ignore the majority?
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Statistically in such cases, the majority is the rule, and the small portion in the middle is the exception to the rule, which should be discarded as being unreliable. If this material were plotted on an x-y graph and a best fit or chi square analysis were run, the center 16 vectors would lie outside the norm and would be discarded. The best fit would be all the vectors that point elsewhere. since there are so many of them. The fact that the minority 16 in the middle are closer to pointing in one region does not inherently make them more relevant, unless you are going to be deliberately biased for that trend, with an agenda in mind, which is, giving undue credence to the flat-earth outlook.
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Besides, those 16 would have to be pointing not to a broad range but to a single point in order to indicate a common direction.
They are not all directed to a single point, so that's further discrepancy against your conclusion.
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You are choosing to observe the minority, at best vague trash or "noise," and discard the vast majority -- the important data.
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Do you have a reason to be so extremely and conspicuously biased in this way?
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The same applies to the arrangement shown at 1:19.

Also, when the lines pass through one another and then diverge away from each other, it makes it look more messy and implausible, as if the light would have to come from multiple directions ... but not if you stop the lines where they converge at the tip of the pyramid-like shape.

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The reason the vectors are shown continuing beyond the region where they seem to converge at 1:46, is for clarity's sake, so their arrowheads can be distinguished from each other and you can see where they're going beyond the apparent convergence. The "messiness" of this situation is a by-product of the data, although from a narrow outlook or one looking for bias, the messiness might seem to be an objective.  
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If the data gets messy would you be more impressed if that messiness were passed over and not mentioned?
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The shape looks more like a prism or a tent than a pyramid, since the top convergence is not a point but a line-like region.
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I said "cone" before but that was before I recognized the depth of the pattern from north to south. If the vectors really did converge at a point the shape would be conical not pyramidal. 
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There is no "point" where the vectors converge. 

I don't see why they would have to converge at a POINT, since the sun would have a much bigger surface area.