“From what I’ve seen, MDiv programs are aimed at training you for (face-to-face) ministerial work.”
Well…that’s true in theory and in the original orientation of many M.Div programs, but not true in practice or usage. I’ve known several people with M.Divs who are not doing ministerial service, and who used them as the basis for academic or scholarly study in religion/theology. For example, Yale Divinity School says of their M.Div: “The degree of Master of Divinity (M.Div.) certifies completion of a program of theological studies designed primarily, although not exclusively, to prepare the candidate for ordination to the Christian ministry.”
And Duke: "The Master of Divinity (M.Div.) program is a three-year degree that prepares students for a wide range of ministries in the church, academy, and world. "
And Harvard:" Graduating MDiv students leave HDS with a toolkit that prepares them to serve within a religious community, to explore the spiritually formative dimension of intellectual work, to evaluate and act on the needs and questions of religious communities and beyond, and to speak with both a pastoral and a public voice."
The last one is particularly relevant to you, I think, since you say that you want to write to a large audience in the style of a theologian. If you want to speak to people of faith (especially Christians) with the voice of someone who is an expert in theology, an M.Div may be especially useful to you because you will learn both the theology AND the pastoral/reaching people part.
However, if you intend to do deep academic research into theology and then write books for popular audiences about it, a ThD may be a decent choice. It’ll teach you to do the research and give you 6-8 years to deep academic theological scholarship. However, that degree is geared towards preparing academic theologians (folks who will go onto teach students in theology at a university), and the writing is going to be largely academic. The way you will be taught to interact with others will also be largely academic. You’re going to have to do some personal work “on the side,” so to speak, to hone your craft to a popular audience.
As for the latter - I don’t know? In my opinion, you’d probably get more acceptance from some and less or no change from others; in the end, it may just be a wash. From my field of psychology, there doesn’t seem to be a strong correlation. One of the most popular writers in it is Malcolm Gladwell, who doesn’t hold any advanced degree at all - much less a doctoral degree. (He also frequently gets the social science wrong, much to the irritation of social scientists.) Simon Sinek is another popular author without an advanced degree in social sciences who writes about psychology (and usually gets it wrong). On the other hand, some other very popular authors who write about psychology - like Steven Pinker, Jonathan Haidt and Claude Steele - are psychologists with PhDs who have done decades of research in the field and write from their wealth of experience. Maria Konnikova is an interesting midway experience - she began writing about psychology after getting a BA in psychology (and studying with Steven Pinker), but later got a PhD in psychology.
So, in my opinion: I don’t think it matters, especially if you would do a master’s degree either way. If your opinions are controversial, you MAY get a little more clout if you do have a PhD and can say that you backed it up with scholarship…but with a popular audience? Eh, maybe not.