I am not in the least surprised North Berwick never hosted an Open, but I am very suprised it never hosted an Amateur.
Here is my take.
The Open was started by Prestwick because they put up the Championship belt and I beleive the money which was introduced the following year to be split among the "runner ups". Once the Belt was given to YTM two other clubs joined in organizing/hosting the Open in the early 1870s; the R&A and HCEG. I would be willing to bet that North Berwick had the opportunity to join this trio of clubs if it were willing to host and put up money for the Claret Jug and prize money. As it was the two leading clubs took over in conjunction with Prestwick. It wasn't for another 20 years or something that another club, Sandwich, got a look into hosting an Open and I would bet there was serious pressure from the London crowd to do so. It just so happens that around this same time, HCEG moved to Muirfield, just a few miles from North Berwick. Once that happened, I don't think there was ever a chance North Berwick was going to host the Open. With the emergence of Hoylake onto the scene as the host of the other major, the Amateur (and eventually the Open in 1987?), more of the power base of golf started to shift south. The coming of the railway which resulted in the stockbroker belt suburbs around London also made the development of the heathands a viable commercial interest and further cemented the role of the south as a major player in British golf. It could well be that the R&A was chosen (early 1920s) to be fully responsible for the running of the Open as a way to balance power between north and south. It also made some sense since the R&A was also responsible for the rules.
It is probably the case that like Prestwick, North Berwick would have struggled to handle the crowds. I don't think the club ever had the presence in North Berwick that the R&A did in St Andrews. Remember, in the early days of the Open, the West Links was not a full 18 hole course. This didn't happen until the late 1870s, but many of the holes were quite short as space didn't allow for anything more. The shape of the present course didn't occur until just shy of the turn of the century. By then, the 9 hole Musselburgh course had been cast aside and Prestwick had long establlished their 18 holer which was still tight for an Open. It would seem that North Berwick never found itself in a position to get involved with the Open and when it did, the times had passed North Berwick by.
Ciao