Actually, it looks like it was bits from YOUR post in 2012 (Save Pages file as html?) that I found and used.
I'm not sure why you say that the styling is lost. Actually, bits of the styling are retained in the HTML that the action creates -- about a dozen 'span' styling elements and another dozen used on the paragraphs, named as generic 'sx' and 'px' styles. But, all in all, the action-generated HTML is decent and conforms to the basics of the very clean (nearly style-free) Word file that was the input.
So, indeed, the automator action works just fine -- better than Word.
Word produces a ton of 'junk' styling, all in conflict with the site's standards. I use Dreamweaver which can strip out some of the offensive 'junk' but a whole lot of of it remains. The HTML that's output from a Word-save is really, really ugly and the webmaster won't accept it as-is. Yes, I can edit in a text editor; yes, I can edit in Dreamweaver (which I what I had been doing). My objective here is to eliminate most of the 'hand work' of the HTML step and get acceptible output from Word, fairly directly.
The Word source is a docx. It uses a standard (minimal and standardized style) Word template to 'normalize' the content from various contributors into one uniform set of conventions. So perhaps pandoc will be even better for my purposes. Thanks for that suggestion.