A less comfortable fact is that of the men-only clubs in the UK, several wield huge influence. There are those that have the high profile of staging the Open, and in the case of the Royal and Ancient there is the added role of governing the game.
Does this not hinder golf as it tries to attract more players and escape its elitist image?
"I truly think that is a myth," Dawson said. "I think many people out there think that the R&A run the golf courses in St Andrews and that no women can play.
"That's absolutely not the case. The courses here, and there are seven of them in the town, are entirely public, run by the St Andrews Links Trust and around these courses there are five private clubs that share the public courses with visitors."
But Dawson's admission that there could be a notion that women aren't welcome at St Andrews does highlight a perception problem.
Leading government figures like Sports Minister Hugh Robertson have described the men-only status of the Royal and Ancient as "increasingly anachronistic".
The issue has disappeared at a stroke for the Augusta National, so why doesn't Dawson encourage members of his own club to follow suit? "I don't deny my job would be made a lot easier if this issue didn't exist, that's self-evident.
"But one might choose to respect the wishes of members of these clubs, which are virtually unanimous in a place like St Andrews, that the status quo works extremely well for them."
Dawson accepts there is growing pressure on high-profile single-sex clubs like the Royal and Ancient to change their ways. "We are very aware of what people say. This issue is really quite small, although quite high profile," he said.
"I think if we have this interview 10 or 15 years from now, we might be talking about quite a different scene. But that's for others to determine."
So within a decade and a half, women might be admitted to clubs like the Royal and Ancient, Muirfield, St George's and Troon? "Well, I think it's possible but I do very strongly believe it is a matter for them to determine.
"I think these pressures quite often misrepresent the facts, but there is no doubt the pressures are there and they are part of the market which will determine what happens here."
Dawson has taken due note of Augusta's move to admit women. "Was it enlightened?" he said. "I suppose you could say that, yes. I wouldn't want to in anyway criticise what Augusta do, they run their own affairs."
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Relations between the R&A and another leading organisation, the PGA of America, are less cordial.
The body that organises the US Ryder Cup effort and represents the country's club pros has been most vocal in opposing the proposed ban on anchored putting methods.
The process has been shrouded in secrecy and the R&A boss believes there should be more transparency in such cases. "This is a PGA Tour matter and I understand that it is being dealt with under their procedure at an appeal stage," he said.
"A determination will be made, I think, quite soon. I can't deny that a little more openness about these things should be encouraged," added Dawson, who helped lead the introduction of drug testing to the game.
The picture appears somewhat clearer regarding the construction of the Olympic golf course in Rio for the 2016 Games. Paperwork problems have prevented the start of building work and the project is a long way behind schedule.
"Just last weekend I'm delighted to say all the permits have been signed by the city of Rio. We do feel there is scope now to get back on track," Dawson revealed.
"It has been a worry for us and we must now make sure the construction process remains on schedule and if it does we will be fine."
And if it doesn't? "The implications are pretty severe because there isn't another golf course that we inspected that would be right for Olympic golf in the city of Rio, but I'm increasingly confident we won't need to have a Plan B."
Dawson reiterated that it is almost certain that 72-hole strokeplay will be used for the men's and women's tournaments at the Olympics.
Both sexes will be represented in Rio but not, for the time being at least, in some of the clubhouses of the game's most iconic venues.
Listen to Iain Carter's interview with Peter Dawson in "5 live Golf" on BBC Radio 5 live at 21:00 BST on Wednesday 24 April
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