I’m still not convinced by either side
That’s absolutely fine; it’s best to inquire into all the arguments and eventually make up your own mind, or simply leave the question open.
As for the weave, it is a little more complicated than you imply. Almost all weave is very simple “tabby” weave. A set of threads is stretched across a frame (the warp threads), and then a shuttle loaded with more thread is woven under and over them (this is the weft thread). It is possible, and in the case of thin strips and some damasks, simply to thread the shuttle like a darning needle, over and under and over and under from one side to the other, and then back again, but this is ridiculously time consuming for large sheets of simple pattern. From very early times the warp threads were arranged so that every alternate thread passed over a bar at the top or bottom of the frame, lifting it up slightly, so that there was a gap between the even (lower) and odd (upper) threads, through which the shuttle could be passed very easily over the evens and under the odds. The clever bit is that all the lower warp threads were tied to a bar across the whole loom, which could be lifted up so that the even threads could be pulled even higher than the odd threads, so that coming back, the shuttle was passed under the evens and over the odds. However, this was still very time consuming, as the bar had to be lifted up onto a support and lowered back to level with the frame with every pass of the shuttle.
There is good evidence from Egyptian records in particular that as looms developed, more horizontal bars were introduced, and the warp threads, numbered 0-1-2-0-1-2 threaded on to two bars rather than one. With this one can make simple twills, by lifting one bar after another as the shuttle was passed backwards and forwards. Naturally, one could project this idea into a loom of more and more bars, but the fact is that there is no evidence for any more than two in the whole pictorial and archaeological corpus.
The four shaft loom is not known until medieval times, by which time the development of looms had resulted in each horizontal bar being attached, via pulleys, to a foot-treadle. By playing the four treadles with the feet, alternate bars can be lifted to produce the 3/1 twill of the Shroud.
There is a discussion of all this, with references, at
The Medieval Weave – The Medieval Shroud.