Gentlemen,
I feel the only way to play the Old Course is to undertake a pilgrimage to Scotland with the starting point being Prestwick, then North Berwick, Musselburgh and finally the Old Course St Andrews.
This is not a race to play as many old GB&I courses as possible, but to savour the whole historical experience as well as to get an understand of each courses over a few days.
Contacts the clubs, make you intention clear, ask for assistance and cooperation from the Members from simply walking the course with hopefully someone who is aware of the history and the changes over the years. Embrace the Membership of each club and you will find it most rewarding. My only advice is to play two round on the same day, repeat it if the weather changes before you move on to the next club/course.
As for the last venue, it would be by far more important certainly for you and your visit to St Andrews that you gain ingress to the R&A Club House, however if that is not possible then contact the New Golf Club St Andrews and use it as your base. They may be able to obtain access to the Trophy Room at The R&A
By allotting more time with fewer courses will IMHO give you far more time to fully understand our golf, our club scene and the pure magic of playing our old courses with support and knowledge of local members.
It will stand you in good stead for future visits to the Home of Golf, and anyway how many can boast that they followed the Great Match Trail.
Like a good wine, we golfers should rest, breath and get to know the country and its heritage before we can fully appreciated the experience. To many courses in 10-14 days just waters down the full enjoyment content and is defiantly not the way to enjoy our Old Courses.
Melvyn