The first issue was that the stitches around the thumb hole were broken because the manufacturer used regular straight stitches with non-stretchy polyester thread. Thumb holes are often stretched when a thumb is inserted and taken out; the result was the stitches popping. Secondly, the sleeve elastic band came undone. Once the chainstitch breaks, the stitches can unravel very easily, which is what happened to this one. I had an easy time taking out the rest of the stitches because all it took was a simple pull on the right part of the thread in the chainstitch.
I thought about using straight stitches with a stretch thread, but I figured that will probably break eventually because the elasticity in the stretch thread is not that much.
In the end, I used zigzag stitches for the repair of the thumb holes. I also used the zigzag stitches on the elastic band around the sleeve for the same reason.
All those choices were made with my client, so we knew what the outcome would be like. In order to make a garment durable, sometimes it cannot look exactly like the original. This is because the original sewing techniques were not very good.
That is perhaps how the large clothing manufacturers ensure that you keep on buying new clothes because they break. I was happy that this client did not give up on his favourite piece of garment.