I have reading this thread with interest, and decided to join this forum. I do not climb in threes, but I have used ropes for climbing in mountains and securing on glasiers. And then I use ropes for some acrobatics and gymnastics use including chock cords. I have read several places concerning ropes for different use.
I did some test with low loads myselves using my own body weight and standing on body scales on the floor. In this way you can measure the force in test items to a fixed point above you.
It have been written here, that the end part of the eye would carry 30% of the load. I can confirm that in tests using steel carabins in the loop.
Then I have tried stitching with a sewing machine. The stiches from one side to the other will typically go about 60 degrees to the direction of the rope. I did try with a few stiches - 2, 4, 6, 10 and 12 stiches. In all cases it seems linear. 12 stiches holds 6 times more than 2 stiches. It will only be half the connecting threads that carries the load. When load is carried then one side of the stiches becommes about 90 degrees to the load and the other half about 15 degrees to the load.
I did also try some hand stitching. In this case almost all treads connecting the sides did carry the load. If you use the mantel only in the stiches, you can pass both ropes in every stich making the double amount of connecting threads on each pass of the needle.
I think hand sewing is good, but I do also have some concerns regarding the core part of the rope. With the stitches you connect very well to the mantel, but it might only be some friction to the core. So in case of few stiches with a larger diameter thread you may not connect to the core sufficiently.
It will also depend on how the core is made. The core for climbing rope have more twisted cordels. So I suppose that helps. For other kind of ropes you have straigt fibres in the core and then its hard for the sewing thread to connect. For double braided rope, the core will normally be braided with a low angle, so it can pass a pennetrading tread a long way before it locks.
The testing for this core problem may also be more difficult and different. I guess you need a lot of loads and unloads of the eye to see if the core starts to creep below the mantel. If it creeps in time, then later on it will only be the mantel strength taking the loads for a heavy fall.
By sewing in hand you may also be able to improve the friction to the core by wiggle the needle while you pass the core. But not to much, because it can break the thread when the rope carry a high load.
I don't say that hand sewing is no good. I would guess it is. But please observe these eyes and the core carefully over time.