I think this is a great question. I'm a professor in the social sciences, so I'll try to keep my advice general and note where I think disciplinary differences might come up. I'm also assuming you're at an American institution.
I agree 100% that Word can be tedious, unstable, and difficult to use - I had similar experiences writing my own dissertation before I had ever learned how to use R. I also think wanting a reproducible dissertation is commendable and it is what we need to see so much more of! @jdlong's advice is really good as is @dano's. Whatever you ultimately choose for writing your dissertation, you need to remember a couple of things.
One, like @jdlong mentioned, you need to focus on content first. I advise my students to start writing in something simple like Byword. Just get the ideas down. The specific tool is less important than having a distraction free space - don't spend time worrying about formatting, figures, etc. Byword is nice because it is a Markdown based editor, but really any distraction free environment will do.
I don't know what disciplinary norms are in civil engineering. In the social sciences, half the chapters in a diss will not have any analyses in them that you would need RMarkdown for (typically). Your intro, lit review, methods, and discussion/conclusion chapters will not likely benefit from RMarkdown as much as other chapters. So unless you really like writing in RStudio, at the minimum find another markdown editor to use as well!
Another thing to keep in mind, which you hint at, is that dissertations live on in a variety of ways. You will use the same analyses for your chapters, publications that come out of those chapters, your defense, job market presentations (assuming you're going the academic route), and conference presentations. Having your analyses so tightly linked to your writing may ultimately mean reinventing the wheel when it comes to converting chapters into other products.
Your committee members are not going to want to get markdown documents, .tex files, or anything similar from you. They will almost certainly want word documents. You can try to show them the light, and there is a lot to criticize about this, but it is the nature of academic writing still and I don't think it is a hill worth dying on for PhD students. Learning pandoc or using knitr to knit to word will be your friend.
The same is true for some journals - they will not accept anything other than word document manuscripts. So keep both of these things in mind - your workflow is going to need to build in room for both of these eventualities. Again, disciplinary boundaries will dictate the extent to which this is true.
Speaking of formatting, you absolutely need to follow @dano's advice in particular about formatting your final dissertation. I do not use RMarkdown templates, so I can't give specific advice here, other than to say that you will have to follow an exacting format. Most institutions have someone designated to check this formatting before you defend. Their job is to make sure it is right, and there usually isn't a lot of flexibility. Ultimately, you will need a .pdf file that you can submit to ProQuest.
Finally, in terms of general advice, the best dissertation is a done dissertation. I do not think that spending a lot of time learning new tools is always the best route for students who need to get job market papers out and defend. I've watched students spend more time learning LaTeX than actually producing a good dissertation. In academia, you will get few bonus points for writing in RMarkdown or using other tools. You absolutely will need a good product (which goes back to @jdlong's point about content).
So what is best? My advice to students is to write their text in an easy to use, distraction free environment. Use plain text so you can track changes using git (@jdlong's advice!). Instead of writing the entire document in R Markdown, focus on having clear analysis R Markdown documents that can be easily tweaked regardless of what their final destination is (chapter, paper, defense presentation, etc). Once you have gone through editing and your committee has signed off on your content, then worry about typesetting. Decide how much time you have and what the best use of it is - learning LaTeX, developing something like what @dano used, or just using Word. Remember - the best dissertation is a done dissertation.
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... I'll try to post this list in a follow-up.