Maybe I need to see this series to see what’s going on! From what you say, it sounds like some of the features of the series are not in themselves entirely implausible, but that the way they are all being brought together is something of a stretch.
For example, Anglican clergy, especially from the Anglo-Catholic wing of the Church, were often at the forefront of race relations. In the 1950s there were definitely Anglican clergy active, and very prominent, in the fight against apartheid in South Africa. At the same time, on the other hand, black immigrants in the UK found a rather mixed reception in parishes. I believe that they found a warm welcome in Anglo-Catholic parishes in the poorest areas of major cities, but many parishes were less welcoming. So while I think it’s entirely plausible that there were Anglican clergy in the 1950s who were far from being racist, it is somewhat unlikely that they were routinely socialising with black people, and in rural East Anglia there simply would not have been that many black people anyway!
Again, there’s been a gay subculture within the Anglo-Catholic movement ever since its inception in the 19th century, so a gay curate in an Anglican parish seems far from unlikely. What does seem unlikely is that he would be meeting other gay men for dates at the cinema. All male homosexuals were vulnerable to blackmail as well as to prosecution, and that would have been all the more of a risk for a clergyman.
I think there’s a tendency in all TV these days to try to introduce as many social issues as possible. You’ll see much the same trend in Call the Midwife. I caught bits of an episode the other day and there was a woman who has just discovered that she is intersex, another woman had an illegal abortion, there was a couple dealing with racism arising from the woman’s decision to marry a Sikh immigrant from India, and a man was suffering Couvade syndrome. A couple of weeks back an elderly woman was revisiting the trauma of force-feeding while imprisoned for campaigning for women’s suffrage. Although these things do happen in real life, it seems that in TV there is an implausibly high concentration of these issues! It can be done well and it can be done badly. Coronation Street does it much better. The issues are developed over a longer period and are woven into regular storylines with a lighter touch.