Do most Anglicans support gay marriage?

The Ozanne Foundation has recently published a piece of research, carried out by YouGov, which claims that 55% of Anglicans believe that same-sex marriage is ‘right’. The resulting headling in The Times was ‘Majority of Church of England worshippers back gay marriage’.
But this is deeply misleading.
In the representative sample used in the survey, 23% of the population of England and Wales said they were ‘Anglican’. That’s 13.5 million people! But we know that the average weekly attendance in Anglican churches is just under 750,000. It is clear, then, that 95% of people who say they are ‘Anglicans’ are not regular church attenders.
Should the Church of England change its doctrine because of the opinions of nominal ‘members’ who seldom, if ever, darken its doors, and are likely to show little interest in, or commitment to, its other beliefs and practices?
It would be better to ask religiously-observant Anglicans about their views on same-sex marriage. But the Oxanne Foundation doesn’t seem interested in that sort of survey.
But, in any case, since when was Christian doctrine determined by opinion poll?
(Based on this article by C of E priest and statistician Peter Ould)