I just did the same as you (tried it again)
and actually managed to slightly improve. The big change is probably in what order you irrigate/road the cattle tiles around Mecca. I irrigated one cattle and then roaded it. Did the same with the second one. I figured that the increased food and growth outweighed getting the increased trade right away. With that, it was warrior/granary/settler/settler/settler. In this second run I tried, I almost managed to end with 5 cities; the four shown in my screenshot up top (same locations too) and a 5th by the game and fish to the east of Medina. I was about 2 turns away from doing it. I'm pretty sure I could hit 5 or 6 if I didn't build the initial warrior in Mecca. Just send one from Medina and use luxuries to keep the peace in the meantime.
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