Mark,
Presumably he's an ex officio member of those clubs.
Are you sure that it is the same P Dawson at the English Golf Union in 1969. There is another Peter Dawson who became a professional golfer and played on the Euro tour.
Here is a short blurb from the R&A when he took over from Bonallack:
"in 1964 Dawson won the Essex boys title. Both (he and Bonallack) were members of Thorpe Hall Golf Club.
After that, Dawson, educated at George Heriots School and Westcliffe High School, graduated from Corpus Christi College in Cambridge with an MA degree in engineering and business studies.
His subsequent career in engineering saw him operate for more than 25 years in senior board positions. A former scratch handicapper - he still plays off three - Dawson is a member of Northumberland Golf Club and joined the R&A five years ago."
So, he is/was a golfer, but his experience is in business, although he was apparently on the R&A rules committee before moving up.
Seems he's turning 65 in 2013. Maybe he'll retire at 65 like his predecessor.