I didn't exclude the rest of your text/quote to be obnoxious. It's just that I'm not sure it added to the discussion.
Yes: some Christian circles advocated against the playing of D&D. Is it relevant that only some did this? I'd answer "no", in that each time a Christian speaks about right or wrong, they're claiming to speak for the Creator Of Everything; they're speaking for God and all true Christians who believe in that God.
The problem has always been that most Christian circles clam-up when a minority view speaks up about what the bible/God have to teach us about <insert some modern topic here>. Most Christian circles refused to get involved when a minority Christian POV started telling people what God wanted them to do about D&D (etc).
This gives normal people the impression that the minority view represents the Christian majority.
And that is the fault of Christianity.
I think the phenomenon you are describing is a trait of human beings in general in a society with a free and open press, that being the idiot who says the most preposterous things the loudest gets all of the media attention and denies quiet and reasonable people the opportunity for reasoned discourse.
American Christians are particularly prone to the problem you describe simply due to the protestant nature of America's founding and the ethos they brought with them (all men created equal, rights come from nature's God, free press, etc. . .). In contrast, the Catholics have the advantage and curse of one person in charge who navigated the politics of the church and is a recognized head of state, i.e. the Pope, who will make less preposterous statements (at least now that he's lost the power to call inquisitions, declare holy wars, and have people murdered who disagree with him over some doctrinal matter). Moving further across the religious spectrum, when atheism is the governing ethos, there is typically no free press, so whatever the guy in charge says is right by default and the press will cover it or bury it as the leader decides and murder people by the millions who disagree with him.
Since a lot of protestant denominations don't recognize a single person being in charge with the exception of Christ and also strongly believe in church autonomy, there is no recognized authority on earth to shut a loudmouthed moron up except his congregation. However his congregation likely self selected to nearly exclusively people who agree with him, and those that disagree with him moved on to some other church resulting in the loudmouth living in an echo chamber (the phenomenon was a sight to behold when I saw it happen).
All of this leaves "normal people" who are mostly Christians (Protestant, Catholic, or otherwise) who just want to live their lives and do the work of Christ bearing the stigma as you say. Since I met most of my gaming buddies in Sunday school, my experience is that Christians by and large are smart enough to not to have listened to the loudmouth on this topic.
I don't know that there is an organizational solution that makes things better in the long term short of Christ's return. History is replete with the crimes of the Popes and it took lots and lots of death and bloodshed for them to relinquish their chokehold on Christianity, so going back to the single leader model is demonstrably a very bad idea. As you say, the church autonomy model leaves something to be desired. The voluntary association model the Southern Baptist Convention is operating with might be a better organizational framework for a somewhat unified Christianity, but would be a tough thing to implement and maintain.
For my part, I'm a firm believer in Jerry Pournelle's Iron Law of Bureaucracy
, I prefer the negatives associated with autonomous churches rather than the negatives associated with the churches aligned under a bureaucracy.